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TANGLE MACHINES I: CONCEPT

DANIEL MOSKOVICH AND AVISHY Y. CARMI

Abstract. Tangle machines are topologically inspired diagrammatic models. Their novel fea-

ture is their natural notion of equivalence. Equivalent tangle machines may differ locally, but

globally they are considered to share the same information content. The goal of tangle machine

equivalence is to provide a context-independent method to select, from among many ways to

perform a task, the ‘best’ way to perform the task. The concept of equivalent tangle machines

is illustrated through examples in which they represent recursive computations, networks of

adiabatic quantum computations, and networks of distributed information processing.

1. Introduction

1.1. The idea in a nutshell. This paper presents a diagrammatic formalism for computation,causation, and information processing. Behind this endeavor is the insight that the combinatorialproperties of knot diagrams mimic principles pertaining to conservation and to manipulation ofinformation in networks.

We construct diagrammatic models called tangle machines, represented by labeled versionsof diagrams such as those of Figure 1, that represent entities and relationships between thoseentities. Unlike labeled graphs, in which edge e from vertex a to vertex b represents a transitionfrom the label of a to the label of b, the basic building block of a tangle machine is an interaction,in which agent c causes a transition from colours of input patients a1, a2, . . . , ak to colours ofcorresponding output patients b1, b2, . . . , bk. A machine makes explicit the cause of a transition.From one perspective, a machine is a computational scheme, a sort of “planar algorithm” whereininteractions simulate basic computations. From the dual perspective, a machine is a networkwithin which information is manipulated at interactions and then transmitted further down toregisters in other interactions. Information can be both a patient (e.g. an input data stream)and an agent (e.g. commands of a computer programme). This aspect of information is capturedby tangle machines but not necessarily by labeled graphs.

The novel feature of tangle machines is their flexibility. Whereas competing graphical modelsare rigid, tangle machines admit a natural local notion of equivalence. Roughly speaking, twomachines are equivalent if one can be perfectly reproduced from the other (the precise definitionis Definition 3.17). As discussed in the sequel paper, machine equivalence parallels the notionof ambient isotopy in low dimensional topology. Local features such as implementation andperformance of computations or information manipulations modeled by the tangle machine may

Date: 10th of April, 2014.

Key words and phrases. diagrammatic models, natural computing, recursion, adiabatic quantum computing,

information theory, cybernetics, networks, knot theory, reidemeister moves.

Figure 1. A tangle machine with colours suppressed.

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http://arxiv.org/abs/1404.2862v1

2 DANIEL MOSKOVICH AND AVISHY Y. CARMI

be different for networks modeled by equivalent machines, but we consider their informationcontent to be the same. We may thus use the tangle machine formalism to select, from amongmany equivalent models which ‘perform the same task’, the model (and thus the network) bestsuited for a specified application. This concept is illustrated in our examples.

The paper is organized as follows. Section 2 explains how the concept of a tangle machineemerges from considerations of low dimensional topology, information, causality, and computa-tion. Section 3 defines machines and machine equivalence. Section 4 presents a diagrammaticcalculus for machines that is similar to the diagrammatic calculus of knot diagrams. In Section 5we illustrate the utility of machines with the following examples:

• Machines representing recursion and Markov chains (Section 5.1).• Machines representing adiabatic quantum computations (Section 5.2).• Machines representing networks of distributed information processing (Section 5.3).

In each example three equivalent machines are presented, one ‘optimal’, one ‘suboptimal’,and one ‘abstract’. This illustrates the operational meaning of machine equivalence.

Appendix A reviews the knot theoretical background to the concept of machines.The sequel to this paper will discuss a machine’s information invariants, and their inter-

pretation in terms of information. Invariants are numbers, polynomials, and other typicallywell-understood mathematical objects associated to equivalence classes of machines. Informa-tion invariants are those invariants v such that, if M1 M2 is the connect sum of M1 with M2,then

(1) v(M1 M2) = v(M1) + v(M2).

Information invariants are topological quantities which encapsulate the information contentof machines.

Further sequels to these papers are planned, and these are briefly discussed in the conclusionto this paper.

The idea to model computations using tangle diagrams is not new. Motivated by (Spencer-Brown, 1969), Kauffman has used knot and tangle diagrams to study automata (Kauffman,1994), nonstandard set theory, and lambda calculus (Kauffman, 1995). Buliga has suggestedrepresenting computations using a diagrammatic calculus involving tangles coloured by quandleswith multiple binary relations (Buliga, 2011b). Farhi et al. have suggested knot diagram equiv-alence as an encryption mechanism for quantum money (Farhi et al., 2012). Roscoe has usedalgebraic structures similar to racks to study computation (Roscoe, 1990). A topological for-malism for interacting processes involving diagrammatic calculus of category theory is discussedin (Baez & Stay, 2011).

1.2. What is a machine? In this section we roughly describe machines via their Reidemeisterdiagrams (Section 4). The precise definition for machines is Definition 3.4.

1.2.1. Interactions. The fundamental building block of a machine is an interaction. The simplest

interaction is graphically depicted as x x ⊲ y

y. This describes initial information x (called

the input) being updated by new information y (called the agent) to obtain updated informationx⊲y (called the output). In general, the updating operation ⊲may differ for different interactions.The labels x, y, and x ⊲ y are called colours, and the strands being coloured represent registers.

Example 1.1. The colours might be invertible operators and the ⊲ operation might be conjuga-tion:

(2) x ⊲ ydef= y−1xy.

We give a specific example of such an interaction. In the traditional circuit model of quantumcomputation, the quantum state is propagated through quantum gates. A quantum gate is aninteraction in which the input is a density operator ρ and the agent is a unitary operator U .The output is ρ ⊲ U , that is a quantum time evolution of ρ.

TANGLE MACHINES I: CONCEPT 3

In general, one register in an interaction may update multiple registers. In this case, the agentis drawn as a thick line. For example (with colours suppressed):

(3)

Updates performed by a single agent should be thought of as simultaneous. But an artifactof the diagrammatic formalism of this introduction (Reidemeister diagrams) is that updatesdo appear to be ordered although in fact they are not (this shortcoming is not shared by thediagrammatic formalism with which machines are defined in Section 3). Thus, the two diagramsbelow, whose diagrams differ by permutation of input-output pairs (on the LHS the agent.indicated by the thick line, appears first to update process A and then process B, while on theRHS it appears first to update process B and then process A), in fact depict the same machine:

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AA

BB

1.2.2. Composing interactions to form machines. A machine is obtained by composing interac-tions, so that the output of one interaction may be the input or the agent of another interaction,and the agent of one interaction may be the input or the output of another interaction. Startingat a point and walking along the machine from input to output and over agents until we arriveback at the point at which we started or at an endpoint of the machine, and in the latter casewalking back until we hit the other endpoint, partitions the machine into components calledprocesses. See Figure 1 for a disconnected example of a machine with colours suppressed.

At first sight, stripped of their colours, machines may look like tangle diagrams in low dimen-sional topology, perhaps like diagrams of w-knotted tangles (Bar-Natan & Dancso, 2013). Infact, as demonstrated in the sequel, diagrams of machines turn out to be diagrams of embeddednetworks of spheres and intervals. Because machines are generated by interactions rather thanby crossings, and because the notion of equivalence of machines reflects this, machines are quitedistinct from the standard fare of knot theory, even without their colours. In addition, thelanguage developed here ascribes a non-topological meaning to such diagrams.

1.2.3. Equivalence of machines. Machines come equipped with a natural local notion of equiv-alence.

First, we care only about the combinatorial pattern in which we concatenate registers, andnot about how we choose to draw the concatenating lines in the plane. This parallels the graph-theoretic distinction between a graph and a graph diagram. Local modifications which switchbetween different diagrams representing the same concatenations, and thus ‘the same’ machines,are called virtual Reidemeister moves. Equation 4 shows another such equivalence, which is anequivalence of Reidemeister diagrams rather than it is an equivalence of machines. The machineson the LHS and RHS of a virtual Reidemeister move and of Equation 4 are considered not merelyto be equivalent, but literally to be the same, with the role of these moves being merely to resolveindeterminacies inherent to the Reidemeister diagram formalism.

Second, the updating operation ⊲ is required not to be lossy, meaning that the input of anyinteraction can uniquely be reconstructed from the agent together with its corresponding output,i.e. if we know the output and the agent, we can uniquely reconstruct the input. This impliesthat ⊲ has an inverse operation ⊳ which we call discounting. The second equivalence, calledR2, states that updating input x by agent y, and then immediately discounting back the sameagent y, is equivalent to doing nothing at all. Of course, in the system being modeled, we mighthave expended energy and resources to update x by y and then more energy and resources to


discount back y (‘equivalent machines are different’), but the computation this performed willhave been the same as doing nothing at all (‘information content is the same’).

Third, updating both the agent and the input, and then updating the updated input by itsupdated agent, gives the same result as updating the output. In terms of the set Q of coloursand its set of binary operations B:

(5) (x ⊲s y) ⊲t z = (x ⊲t z) ⊲s (y ⊲t z) ∀x, y, z ∈ Q ∀ ⊲s, ⊲t ∈ B.

A machine in which an agent updates all outputs of an interaction is equivalent to a machinein which that same agent updates all inputs of that same interaction, and also updates its agent.This equivalence is called R3.

Machines do not ascribe physical meaning to colours, but only to differences between colours.Thus, machines which differ by an automorphism of the rack of colours (Q,B) on one of theirsplit components are considered equivalent.

Finally, for a class of machines called quandle machines, if the input register coincides with theagent (dually, if the output register coincides with the agent), then the computation performedis trivial. Thus, actions of registers on themselves can be added to and removed from machinesat will. This equivalence is called R1. All of our examples in Section 5 are quandle machines.

An agent with no patients (an empty interaction) does not perform a computation, althoughsuch an agent does represent a physical component of the network. To add or delete emptyinteractions from a machine is a stabilization move (ST), and the ST move is called a stableequivalence (but not an equivalence).

Figure 2 lists all virtual Reidemeister moves, Reidemeister moves (R1, R2, and R3), theUC move of 4, and stabilization. In this figure and in the future, a strand drawn without anorientation signifies a strand whose orientation can be chosen arbitrarily.

Why is our notion of equivalence based on this set of moves and not others? Firstly, in thesequel, a Reidemeister Theorem for machines is proven, which shows that certain topologicalembedded objects modeled by colour-suppressed machines are ambient isotopic if and only ifthe corresponding machines are equivalent. So there is a sense in which equivalent machinesare those machines that ‘are entangled in the same way’. Secondly and more intrinsically, itseems that these moves are the universal set of local equivalences for machines. A local moveis a modification of a machine which replaces one fixed configuration of interactions N in amachine M , in a disc D ⊂ R2, by a different fixed configuration of interactions N ′ in D. A localequivalence is a local move on a colour-suppressed machine which is valid for any colouring ofthe machine.

Conjecture 1.2. Any local equivalence on machines is a finite combination of the moves ofFigure 2.

We give an operational meaning to equivalence. For concreteness, suppose for example thatM was designed to solve a computational task with initial conditions encoded by the colours inits input registers, and with the result of its computation being the colours in its output registers.The sets of input and output colours in M coincide for equivalent machines. A machine M ′

that is equivalent to M has input and output colours identical to those of M , but it performsentirely different computations in between (although the computations of M can be perfectlyreconstructed from the computations of M ′, and vice versa). In this way, M and M ′ are twodifferent but equivalent schemes for solving the same computational problem. Metaphorically,equivalent machines represent the same computational scheme written in different languages.Information invariants of machines (discussed in the sequel) are ‘language independent’, and wethink of them as measures of the information content of a machine, in particular as measures ofa machine’s complexity.

2. Conceptual underpinnings

2.1. Machines and causality. A process embodies a flow of data. This metaphorically sug-gests that processes in a machine encode a direction as well as an order of the colours which are


VR1

VR2

VR3

SV

R1

R2

R3 UC

ST

Figure 2. Local moves for machines, valid for any orientations of the strands.The R1 move is valid for quandle machines but not for rack machines.

stored in the registers along it. For a tangle machine, the data along a process is a sequenceof colours, infinitely repeating if the process is closed (see Figure 3). Note that we have madeno mention of physical time. Indeed, what we call the ‘direction’ of a process may conform tophysical directionality or may be opposite to it. Nevertheless, a causal structure exists locallyfor each interaction, and is manifested as the distinction between inputs and outputs.

The idea of causality without an arrow of time harmonizes with a similar concept in the theoryof probabilistic causality, wherein causation is valid only in the presence of a manipulating factor(for a machine, the manipulating factor is the agent register). This perspective gives a causalmeaning to logical statements such as a→ (b→ c), which is interpreted not merely as the usualmaterial implication (if-then connective) but rather as the statement “b transforms to c underthe influence of a”. This is well adapted to describe an abstract computation (without a physicalrealization), namely, “b (input) is acted upon by a (also an input) to yield c (output)”. Slightlyrearranging while disambiguating the roles of the two inputs (one of which acts whereas the


other is acted upon) we uncurry a→ (b→ c) to get (a, b)→ c or otherwise a crossingb c

a,

which is in itself an interaction or a part thereof.

aa

a

aa

a

bb

bb

b

cc

cc

c

M0 M1

Figure 3. A single-process closed machine. The data flow in the process M0

emerges upon tracing the colours (a, b, c) incident to each patient register in M1.

An interaction is by nature a symmetry breaking mechanism. The two classes of patientregisters, inputs and outputs, are very much distinct. The agent register may be viewed as awall which separates the “past” patient from its “future” embodiment as an output register.Of course, the register itself does not undergo any transformation, its only the content whichtransforms by the aid of an agent register. Yet the idea is clear. Causal order takes the place oftemporal order, providing us with nearly the necessary justification for calling a concatenationof registers a process.

At first glance it seems as though we are confronted with a logical clash: how can the causaland temporal orders not stem from one another? Pearl gives an account of a related perplexityin the foundations of probabilistic causality. He notes that “determining the direction of causalinfluences from nontemporal data raises some interesting philosophical questions about the re-lationships between time and causal explanations” ((Pearl, 2009), page 57). The essence of theargument presented by Pearl is that, in a statistical framework such as probabilistic causal-ity, causal relationship between two variables may be stipulated under the influence of a thirdvariable, as echoed in the paradigm “no causation without manipulation” (Holland, 1986). Thelanguage of describing causal relations transcends the physical domain and belongs instead tothe logical domain. We may inquire about causal influences based purely on conditional depen-dencies without reference to physical time. Following this rationale, we assert that an interactionis most profoundly a logical statement of the form:

Agent (the cause) manipulates the relationship (the effect):input content → output content.

A process in a tangle machine can be viewed as a channel through which causal effectspropagate under the influence of various agents. This causal order (flow) is what makes up thecomputation of the (logical) process. Machines avoid direct reference to linear (physical) time.

2.2. Computation. One way to think of an interaction is as a computation. For example, givencolours for the input register and for the agent, compute the colour for the output register. Theagent is not affected by this operation. Example 1.1 realizes a quantum gate as an interaction.

There are a number of computational paradigms that may thus be realized by machines.

Remark 2.1. Kauffman has described a formalism for studying automata using coloured knots(Kauffman, 1994). Reidemeister moves are described simply as ‘the rules of the game’, and theinterpretation of the diagrams is in terms of multi-valued logics, which is different from whatwe do.

2.2.1. Recursion. By nesting copies of the same machine inside one another, machines mayrepresent recursive computations and Markov chains. We exhibit three machines, one of whichhas a stochastic transition matrix, and two of which of which have a stochastic two-step transition


matrix but not a one-step transition matrix. One of these represents a feed-forward system, whilethe other represents a feedback loop. This is discussed in Section 5.1.

2.2.2. Adiabatic quantum computation. The formalism of adiabatic quantum computation isparticularly well suited to being modeled by machines. Equivalent machines may represent net-works of adiabatic quantum computations with different energy gaps, and therefore machinescan provide a means to describe speedups of adiabatic quantum computations which are actu-ally programme independent. Adiabatic quantum speedups will be discussed in future work.Section 5.2 discusses how machines can represent networks of adiabatic quantum computations,and presents equivalent machines which perform the same computations, but with quite differentenergy gaps.

2.2.3. Information. Taking colours to be entropies, the machines of Section 5.3 represent dis-tributed information processing. The difference between input and output entropies representsthe capacity of the computation, and the computation is said to be optimal if this number equalsthe mutual information of the input and the output. We exhibit three equivalent machines whichrepresent computations with are locally optimal, locally suboptimal, and abstract. This para-digm, which we plan to study further in the future, takes the formalism of tangle machines intothe realm of information theory.

2.2.4. Colour processor. Consider (A)lice, who has a machine, transmitting information to(B)ob, who knows only the underlying graph of the machine and the interactions, but whodoes not know the colours in its registers. Alice sends messages to Bob such as “Register r27has colour x”. Based on such information, Bob would like to compute the colour of a specificregister s, or of a group of registers. Thus Bob performs a computation whose inputs are themessages sent by Alice and whose output is the set of possible values for s.

Using the example of classical quantum gates (Example 1.1), we emphasize that a colourprocessor performs reversible computations. Let U be a unitary operator and let ρ be a densityoperator. Then the computation captured by a single crossing is ρ ⊲ U . Reversibility may thenbe expressed as simply (ρ ⊲ U) ⊳ U = ρ, diagrammatically expressed as follows:

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ρ

ρ

U

ρ ⊲ U

Remark 2.2. Machines are more flexible than the circuit model. Thus we can have densityoperators acting on one another and even on other unitary operators. It might be unwise to tryto interpret such interactions in terms of the circuit model. Our setting is different.

2.2.5. Machine as computer programme. An archetypal setting of causation without time is acomputer programme. A computer programme executes in time, but the instructions themselvesare independent of time.

Consider colours as operators acting on a space. For example, colours might be elements of a

group G and the quandle operation might be conjugation g ⊲ hdef= h−1gh, and we might specify

a representation of the group as linear transformations of a vector space V over a field F , via agroup homomorphism Φ: G→ GL(V, F ).

Feed a vector v ∈ V into a chosen input register. When v crosses under an agent coloured g,transform v either to (Φg)(v) or to (Φg)−1(v) depending on the direction of the agent. When vcrosses over a patient, transform v either to (Φg)−1(v) or to (Φg)(v) depending on the directionof the agent. Compare (Fenn, 2012).


(a) LHS of R2. (b) RHS of R2. (c) Decomposed RHS of R2

Figure 4. Subfigures 4a and 4b are trivial, but Subfigure 4c is not. Subfigure4c is a valid splitting of Subfigure 4b as a virtual tangle and as a w-tangle, butnot as a machine.

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g v w h

(Φh)−1(w)

(Φh)(v)

g ⊲ h

g v w h

(Φh)(v)

(Φh)−1(w)

g ⊳ h

The value of the vector at a chosen output register is the computation of the machine.

2.3. Low dimensional topology. Our diagrammatic model of machines as concatenated in-teractions is inspired by formally-similar structures which occur in low dimensional topology.Combinatorial knot theory studies knots as planar diagrams instead of as embedded objects in3–space. These diagrams are decomposed into tangles (Conway, 1970). Knots and tangles aremodified by local moves, which replace one tangle within a knot by another. Knots are thusrevealed to be algebraic objects arising as concatenations of crossings (which are very simpletangles) in the plane (Jones, 1999).

The combinatorial paradigm of knot theory manifests a new philosophy of what constitutesalgebra. For the combinatorial knot theorist, algebra no longer consists merely of formal ma-nipulations of strings of symbols, but rather of operations and local modifications of labeledfigures in the plane and in higher dimensions. This new philosophy of diagrammatic algebra hasbecome particularly well established in the representation theory of quantum groups, in highercategory theory, and in quantum field theory.

Dropping the requirement that concatenation be planar, Kauffman defined virtual tangles(Kauffman, 1999). A natural strengthening of the equivalence relation imposed on virtual tanglesgives rise to w-tangles. Our diagrammatic calculus is most directly motivated by Bar-Natan andDancso’s diagrammatic calculus of w-tangles (Bar-Natan & Dancso, 2013). These formalismsare recalled in the appendix.

The major differences between machines and virtual tangles or w-tangles are firstly thatmachines by default are coloured, and secondly that a register in a machine is an ‘unsplittableatom’. There is no nontrivial way to slice the ‘R2 machine’ of Equation 6, for example. Butthere is no restriction on how virtual tangles or w-tangles may be split. Thus, a virtual tanglewithout crossings is equivalent to a virtual tangle that can be split into a collection of crossings.The same is true for a w-tangle without crossings. This is illustrated in Figure 4. As a result,machines have a well-defined complexity invariant, discussed in the sequel to this paper, whichis roughly the maximal number of ‘non-unit machines’ (machines which represent nontrivialcomputations) into which the machine may be split. But Figure 4 illustrates that there is no‘complexity’ invariant for virtual tangles or for w-tangles, even if we colour them. Indeed, ifwe were to assume that such an invariant exited, then a tangle without crossings which hascomplexity 0 could be split into two tangles with complexity 1 (the crossings) and therefore hascomplexity 2, so 0 = 2, which is a contradiction.


3. Machines and machine equivalence

This section forms the heart of the paper. In Section 3.2 we define machines. These areanalogous to Gauß diagrams in knot theory, defined in Section A.4. In Section 3.3 we constructmachines as a higher-dimensional algebra, in the vein of Section A.8. Finally, in Section 3.4 wediscuss Reidemeister moves and equivalence for machines.

3.1. Interactions and conservation laws. A machine (Definitions 3.4) consists of registerswhich contain elements of a set of colours Q (Definition 3.2). A transition between coloursx, y ∈ Q caused by an agent r coloured z ∈ Q is illustrated by a diagram such as

(8)x ⊲r y

z

This transition underlies an algebraic relation of the form y = x ⊲r z, where ⊲r : Q×Q→ Qdenotes a binary operation associated to the register r. Alternatively we may view the diagramas representing a right action ⊲r of an agent z on an input x to yield an output y. Such pictures,called interactions, are the basic building blocks of our theory (Definition 3.6). We suppress thesubscript ‘r’ in figures of the form of 8 for simplicity when there is hopefully no risk of confusion.

Let B be a set of binary operations on Q. The relations in (Q,B) are referred to as the lawsof the machine. The colours satisfy a universal conservation law stating that

⊲ z : Q → Q

x 7→ x ⊲ z

is an automorphism of (Q,B) for all z ∈ Q and for all ⊲ ∈ B. This condition splits into twosub-conditions.

Law 1 (Global Conservation). The map ⊲ z is an isomorphism of sets. In particular it has aninverse, ⊳ z.

Law 2 (Local Conservation). The right action via ⊲s of x on y gives b if and only if theright action via ⊲s of y ⊲t z on x ⊲t z gives b ⊲t z where ⊲s, ⊲t ∈ B. In other words,

(Distributivity) (x ⊲s y) ⊲t z = (x ⊲t z) ⊲s (y ⊲t z) ∀x, y, z ∈ Q ∀ ⊲s, ⊲t ∈ B.

In diagrams, Local Conservation tells us that

(9)x ⊲ y

z

if and only if x ⊲s b ⊲ y ⊲s b

z ⊲s b

for all b ∈ Q. The left diagram tells us that y = x ⊲r z, and the right diagram tells us that(x ⊲s b) ⊲r (z ⊲s b) = y ⊲r b, yielding Distributivity.

Example 3.1. For the operation x ⊲ ydef= y−1xy for all registers, we have:

(10) (x ⊲ z) ⊲ (y ⊲ z) = z−1y−1zz−1xzz−1yz = z−1y−1xyz = (x ⊲ y) ⊲ z.

An important special case is when B is a single element set, i.e. ⊲r = ⊲s for any two registersr and s in M . In this case, laws 1 and 2 underly a structure known as a rack or an automorphicset. In this paper, we extend the word ‘rack’ to cover the case of multiple different binaryoperations, each associated to a register. Racks which satisfy the following additional conditionare called quandles.

Law 3 (Idempotence). The equality x ⊲ x = x holds for all colours x in Q and for all ⊲ ∈ B.

The above discussion assembles to Definition 3.2.

Definition 3.2 (Rack; Quandle).


• A rack, or an automorphic set, is a set Q equipped with a set B of binary operationssuch that ⊲ z is an automorphism for all z ∈ Q and for all ⊲ ∈ B, and such that Q isclosed under the inverse operation ⊳ of ⊲.• A rack all of whose colours are idempotent with respect to all ⊲ is called a quandle.

Example 3.3. Buliga has defined a distributive Γ–idempotent right quasigroup (Buliga, 2009,2011a). This is an example of a quandle with multiple binary operations, in which elements ofB are indexed by a commutative group Γ, and the following relation is satisfied:

(11) x ⊲s (x ⊲t y) = x ⊲st y ∀x, y ∈ Q ∀ s, t ∈ Γ.

3.2. Tangle machines. Tangle machines are obtained by concatenating interactions.

Definition 3.4 (Tangle machines). A tangle machine M is a triple Mdef= (G,φ,ρ) consisting

of:

• A disjoint union of directed path graphs A1, . . . , Ak ( open processes) and directed cyclesC1, . . . , Cl ( closed processes),

(12) Gdef= (A1

∐

A2∐ · · · ∐ Ak)

∐

(C1∐

C2∐ · · · ∐ Cl) ,

The graph G is called the underlying graph of M . Vertices of G are called registers.• A partially-defined interaction function

(13) φdef= (φ, sgn): E(G)→ V (G)× {+,−}

• A colouring function ρdef= (ρ, ) from V (G) to a rack (Q,B) such that, if v and w are

registers in M and if e is an edge from v to w, we have:

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ρ(v) ⊲φ(e) ρ(φ(e)) = ρ(w), If sgn(e) = +;ρ(v) ⊳φ(e) ρ(φ(e)) = ρ(w), if sgn(e) = −;ρ(v) = ρ(w) if e /∈ Domain(φ).

where ⊲φ(e) is shorthand for (φ(e)), the binary operation associated to the register φ(e).

Remark 3.5. Although one may consider infinite machines, all machines in this paper are as-sumed to be finite, by which we mean that their underlying graphs are finite.

Before reading further, the reader may wish to glance at diagrams of machines on the nextfew pages in order to digest the above definition.

We graphically represent φ by drawing dashed arcs from oriented edges e ∈ E(M) to regis-ters φ(e) ∈ V (M). We label the edge ⊲ if sgn(e) = +, i.e. if the output is the input right-actedon by the agent, and ⊳ if sgn(e) = −, i.e. if the input is the output right-acted on by the agent.Thus, we write

(15)x ⊲ x ⊲ y

y

and x ⊲ y ⊳ x

y

To simplify notation, let ⋄ denote either ⊳ or ⊲. We name various parts of machines:

Definition 3.6 (Interaction; Agent and patient registers). For a register y, write φ−1(y)def=

{e1, . . . , ek} and denote the predecessor of ei by xi for i = 1, 2 . . . , k (not excluding the casek = 0). An interaction is a machine of the form:

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x1 ⋄ x1 ⋄ y x2 ⋄ x2 ⋄ y · · · xk ⋄ xk ⋄ y

y

A register in the image of φ is called an agent (because it ‘acts’), whereas an input register ofan operation is called a patient (because it is ‘acted upon’). In the picture above, y is the agentregister whereas x1, x2, . . . , xk are the patient registers.


We illustrate Definition 3.6 with an example. A register coloured x whose φ–preimage is asingle edge is of one of the forms:

(17)x ⋄ z

y

or x ⋄ y or x ⋄ x or x ⋄

Another definition:

Definition 3.7 (Initial and terminal registers; endpoints). Initial registers are registers whosein-degree is zero. Terminal registers are registers whose out-degree is zero. Initial and terminalregisters are collectively called endpoints.

In general, a machine consists of multiple interacting processes, i.e. registers of one processmay feed operators of another process.

Definition 3.8 (Rack and quandle machines). A machine M whose registers contain coloursin a quandle Q is called a quandle machine. Otherwise, usually when we want to stress that Qis not a quandle, M is said to be a rack machine.

Our first examples of machines are also definitions:

Definition 3.9 (Various degenerate machines).

• A machine with no registers is called a null machine.• A machine with no edges is called an empty machine.• A machine with no interactions (i.e. Domain(φ) = ∅) is trivial.• A machine with one colour (i.e. ρ(r) = ρ(r′) for any two registers r and r′) is a unit.

Two more examples of machines are given below. Further examples are scattered throughoutthe paper.

Example 3.10. The following machine has two open processes.

(18)

x ⊲ x ⊲ (y ⊲ x)

y ⊲ y ⊲ x ⊲ (y ⊲ x) ⊲ (x ⊲ (y ⊲ x))

Example 3.11. The machine below has one closed process, and a ⊲ b = b ⊳ a for all a, b ∈ Q.

(19)

⊲ y y ⊲

x ⊲ y y ⊲ x

⊲ x ⊲ y y ⊲ x ⊲

3.3. New machines from old. In this section we describe some simple ways in which machinesmay be constructed from one another.

(i) Disjoint union: The disjoint union of two machines M and N is a machine M∐

N . Adiagram for M

∐

N is obtained by placing diagrams of M and N side by side in theplane.

(ii) Deletion: Deleting a split component N of a machine M gives a machine M \ N . Inother words, deletion replaces a machine which can be written as M

∐

N either by Mor by N .

(iii) Concatenation: If an initial register a in a machine holds the same colour x as a terminalregister b in the machine, then adding an edge e between a and b, with e /∈ Domain(φ),gives a new machine.


(iv) Splitting : The inverse of concatenation. Deleting an edge gives a new machine.

(v) Connect sum: A machine Mdef= (G,φ,ρ) is a connect sum of M1

def= (G,φ1,ρ1) and

M2def= (G,φ2,ρ2) if, writing A1

def= Domain(φ1) and A2

def= Domain(φ2), we have

A1∩A2 = ∅ and Domain(φ) = A1∪A2, with ρ(r) = ρ1(r) for all r ∈ A1 and ρ(r) = ρ2(r)

for all r ∈ A2. In this case we write Mdef= M1 M2.

(20)

⊲ y y ⊲

x ⊲ y y

⊲ x ⊲ y y ⊲ y ⊲

⊲ y y ⊲

y y ⊲ x

⊲ y ⊲ y y ⊲ x ⊲

=

⊲ y y ⊲

x ⊲ y y ⊲ x

⊲ x ⊲ y y ⊲ x ⊲

If all trivial machines with the same underlying graph are equivalent (a conditionon the rack (Q,B)), then ‘connect sum’ is a well-defined operation. In this case, theset of machines with fixed underlying graph G forms a commutative monoid under theconnect sum operation. The identity element is the unique trivial machine on G.

The converse of connect sum is cancellation. To cancel a factor N = (H,φH ,ρH)

in M = (G,φ,ρ) is to replace M by a machine M − Ndef= (G,φG−H ,ρH) where ρH

satisfies ρH(r) = ρ(r) for all r ∈ G−H.

(21)

⊲ y y ⊲

x ⊲ y y ⊲ x

⊲ x ⊲ y y ⊲ x ⊲

cancel N−→⊲ y y ⊲

y y ⊲ x

⊲ y ⊲ y y ⊲ x ⊲︸︷︷︸

N

We observe the following:

Proposition 3.12. Any machine can be concatenated out of a disjoint union of interactions.

Example 3.13. Consider the following machines, which we call M and M .

(22) (M)

x ⊲ y

y ⊲ y ⊲ x ⊲ x

x ⊲ y

x ⊲ y ⊲ x ⊲ y

(M)

The initial register in one process in the machine M has the same colour as the terminal registerin the other process, and vice verse. Identifying these registers in pairs, we obtain a new machineequivalent to M , having a single closed process. Machine M was obtained from machine M byconcatenation, while machine M could have been obtained from machine M by splitting.

Definition 3.14. A machine that can be obtained as a disjoint union of two non-null machinesis said to be split.

3.4. Equivalence of machines. In this section we describe when two machines should beconsidered equivalent.

The following local modification, in which the top central register is outside the image of φ,is an equivalence.

(23)x ⊲ x ⊲ y ⊳ x

y

R2←→x x x

y

R2←→x ⊳ x ⊳ y ⊲ x

y


For example:

(24) (M)

⋄ ⋄ ⋄

x ⊲ x ⊲ y ⊳ x

y

R2←→

⋄ ⋄ ⋄

x x x

y

(M ′)

Law 1 guarantees that (x ⊲ y) ⊳ y = x for all ⊲ ∈ B so that M and and M ′ are consideredequivalent. This local move is called Reidemeister II or R2 because it parallels the secondReidemeister move for knot diagrams as discussed in Section 4.

Third, the expression of local conservation (Law 2) is called Reidemeister III or R3. It takesthe form:

(25)

(x1 ⊲ z) ⊲ (y ⊲ z) · · · (xk ⊲ z) ⊲ (y ⊲ z)

⊲ y ⊲ z ⊲

x1 ⊲ z ⊲ y xk ⊲ z

⊲ z ⊲

x1 · · · xk

R3←→

(x1 ⊲ y) ⊲ z · · · (xk ⊲ y) ⊲ z

⊲ y ⊲

x1 ⊲ y ⊲ y ⊲ z xk ⊲ y

⊲ z ⊲

x1 · · · xk

The relation should hold for all k = 0, 1, 2, . . . and for all orientations, i.e. if all ⊲ operationsof z are changed to ⊳ operations, or if all ⊲ operations of y are changed to ⊳ operations. Registerz, but not register y, may lie in the φ–image of additional edges not appearing in the figure.Although not explicit in the figures, operations in B associated to different registers via the map may be different.

Example 3.15. The R3 move for k = 0 reads:

(26)y ⊲ y ⊲ z y ⊲ z

z

R3←→y y ⊲ y ⊲ z

z

Example 3.16. The R3 move for k = 1 reads:

(27)

x ⊲ x ⊲ y ⊲ (x ⊲ y) ⊲ z

z

y ⊲ y ⊲ z

R3←→

x ⊲ x ⊲ z ⊲ (x ⊲ z) ⊲ (y ⊲ z)

z

y ⊲ y ⊲ z

In Section 4 we will show that the R3 move may equivalently be drawn, in a diagrammaticlanguage described there, as:

(28)

If Q is a quandle then we impose an additional equivalence relation:

(29) x ⋄ R1←→ x and xR1←→ ⋄ x


Secondly, composing a colouring function ρ : V (G)→ (Q,B) of a machine M with underlyinggraph G with an automorphism α : (Q,B)→ (Q,B) gives a new machine M ′ which we considerequivalent to M . This is a global equivalence. More generally, if M = M1

∐

M2, then thecomposition of the restriction to M1 of ρ with α, together with the restriction of ρ to M2, is alsocalled a global equivalence.

Finally, stabilization subdivides a register into two registers with the same colour, and desta-bilization unites two such registers, giving a new machine M ′ which we consider not-quite-equivalent, but rather stably equivalent to M .

(30) x x ←→ x

After stabilization, it is required that at least one of the two registers be outside the imageof φ. The following are valid stabilizations:

(31)⋄ ⋄

x x

←→⋄ ⋄

x

←→⋄ ⋄

x x

We may now define machine equivalence.

Definition 3.17 (Equivalent machines). Two rack machines (quandle machines) are said to beequivalent if one can be obtained from the other by global equivalence, R2, and R3 moves (andR1 moves). They are stably equivalent if we also also require at least one stabilization and/ordestabilization to obtain one machine from the other.

Example 3.18. The following two machines, which are coloured by a rack for which a ⊲ b =b ⊳ a for all a, b ∈ Q, are stably equivalent as they are obtained from one another by R2 and(de)stabilization.

(32)x ⊲ y

y ⊲ y ⊲ x ⊲ x

x ⊲ x ⊲ y ⊲ y

y ⊲ y ⊲ (x ⊲ y) ⊲ x ⊲ y ⊳ x

Example 3.19. Consider the following two machines:

(33)

⊲

x y

⊲ y ⊲ x ⊲

⊲

y x

⊲ x ⊲ y ⊲

These two machines are equivalent via an automorphism of (Q, ⊲) that exchanges x with y.

4. Reidemeister diagram of a machine

In this section we draw machines in a way reminiscent of the Reidemeister diagrams of knottheory. The construction mimics the algebraic definition of tangles in Section A.8. This section isformulated in the language of tangle machines for simplicity, but its constructions all generalizein a straightforward way to tangled graph machines.

Definition 4.1 (A crossing). A crossing is a thickened strand called an over-strand which endsin two thin strands pointing to and from a firmament (a disjoint collection of circles in R2),which over-crosses k ∈ N thin strands which pass under it either up or down. If k = 0 then acrossing is called an empty crossing and the over-crossing ‘crosses over the empty set’. Endpointsof thin strands are on a firmament.


Example 4.2. The following crossing has three under-strands pointing up, and one pointingdown.

(34)

Definition 4.3 (A machine tangle). A machine tangle is defined recursively:

• Crossings and the empty tangle are machine tangles.• A disjoint union of two machine tangles (placing one besides the other) is a machinetangle.• Concatenation in a machine tangle T connects a stud in T (an ‘out’ endpoint of thetangle on the firmament) with a hole in T (an ‘in’ endpoint of the tangle on the firma-ment) by an edge in the plane, and connects their supports by regular neighbourhoods ofthe connecting edge. A concatenation of a machine tangle is a machine tangle.

A Reidemeister diagram of a machine M is a coloured machine tangle obtained recursivelyby replacing each register r by a crossing, in which the over-strand represents the agent registerand the under-strands represent the edges φ−1(r):

(35)

x1 ⋄ x1 ⋄ y x2 ⋄ x2 ⋄ y · · · xk ⋄ xk ⋄ y

y x1 x2 xk

The under-strands are drawn to point up if sgn(e) = + for the corresponding edge in M , andto point down if sgn(e) = −.

For aesthetic reasons, we sometimes omit to thicken the strand corresponding to an agentwith one input and one output.

Concatenation of registers a and b in machine M corresponds to drawing an edge to connectthe corresponding endpoints of strands, combining the strands representing a and b into a singlestrand. Colours are written beside the corresponding thin strands (for machines, the coloursactually sit on the half-edges). Below are a few examples of machines where a⊲b = b⊳ a togetherwith their respective Reidemeister diagrams:

(36)y

y

xx

⊳ y ⊲

x x

⊳ y ⊳ x ⊳ y ⊲ x ⊲ y ⊲

(37)y

y

xx

⊳ y y ⊲

x x

⊳ y ⊳ x ⊳ y ⊲ x ⊲ y ⊲


(38)y

y

xx

⊳ y y ⊲

x x

⊳ y ⊳ x ⊳ y ⊲ x ⊲ y ⊲

(39)

x21

x24

x11

x13

x13 ⊲ x12 : ⊲ x22 ⊲

⊳ ⊳ x23 x21

x14 x11 ⊲ x24 ⊲

⊳ x15 ⊳

The Reidemeister diagram of a machine is not unique. We define two Reidemeister diagramsto be equivalent if they may result from the same machine, in other words if they differ by:

(i) Placement of lines used to concatenate.

(40)

VR1

VR2

VR3

SV

(ii) Placement of crossings. Shuffling crossings relative to one another in the plane gives adifferent diagram of the same machine.

(iii) Permutation of ‘under-strands’ of a crossing. A switch of under-strands alters thediagram as follows:

(41) i.e.UC

In other words, Reidemeister diagrams are to machines as graph drawings are to combinatorialgraphs. Reidemeister diagrams generalize in a straightforward way to diagrams for tangled graphmachines. The R2 and R3 moves on machines correspond to the following moves on Reidemeisterdiagrams (colours suppressed):


(42)

R2

R3

If R is a quandle, then we also have the R1 move:(43)

R1

Stabilization corresponds to the following move on a Reidemeister diagram:(44)

To translate from a Reidemeister diagram back to a machine, stabilize the Reidemeisterdiagram until there is at most one under-strand between every pair of over-strands, i.e. movingfrom an over-strand down a thin line, we cross under at most one other over-strand before theline again thickens into the next over-strand. Next, map each over-strand to a register. For eachunder-strand connect the two incident registers by an edge and concatenate as required. Recoverφ by joining the register representing the over-strands with edges representing under-strands bydotted lines, and recover sgn from the direction of the under-strand, judged according to theright-hand convention.

The following proposition follows from the construction. As discussed above, stabilization isstrictly necessary.

Proposition 4.4. Stable equivalence classes of machines bijectively correspond to the set oftheir Reidemeister diagrams modulo the appropriate Reidemeister moves, global equivalence andinflation.

It is instructive to point out a number of ‘reasonable looking’, but forbidden, irreversible ornon-conserving moves. These are shown in Figure 5.

5. Examples

This section provides three examples of machines representing computational schemes. Eachexample represents a distinct computational paradigm, and will be discussed in greater detailin future work. In Section 5.1, the machines represent recursive computations and Markov pro-cesses. In Section 5.2 machines represent adiabatic quantum computations. And in Section 5.3,machines represent networks of information processing.


forbidden R2

forbidden R3

forbidden UC

Figure 5. Example forbidden moves. The moves drawn above are not equiva-lences of machines, because they do not have unique inverses. Thus, permutingunder-strands of a crossing induces an equivalence, but permuting over-strandsin different crossings does not.

Throughout this section we make use of the fact that equivalent machines all have the sameinitial colours and the same terminal colours. Thus, the set of initial colours of a machine, andthe set of its terminal colours, are both machine invariants.

Equivalence of machines is demonstrated in each example. In each case, equivalent machinesperform the same computational task ‘globally’, but have different local properties which wouldlead to preferring one over another. One of the equivalent machines in each case is ‘optimal’,another ‘suboptimal’, and a third ‘abstract’, where the meaning of these terms is different ineach example.

5.1. Recursions and nesting. Recursion lies at the heart of computational paradigms suchas automata and Turing machines. It manifests the principle that the future state is determinedexclusively by the current state and by recent input. Similar concepts underlie several widelyused probabilistic models such as Markov chains and autoregressive processes.

To realize recursion in a machine, consider copies M0,M1,M2, . . . of a fixed machine M . LetIn(Mi) and Out(Mi) denote the initial and terminal registers of Mi respectively. We refer toregisters of closed processes in M as control registers. Write U(M) for the set of control registersin M .

Initialize the registers of In(M0) to the initial state of the recursion, and initialize also thecontrol registers within each Mi. For each process P of M , concatenate the terminal register ofthe copy of P in Mi with the initial register in the copy of P in Mi+1, for i = 0, 1, 2, . . .. Denotethe resulting machine M . For each i = 0, 1, 2, . . ., the result of the computation of Mi appearsas the colours stored in Out(Mi), assuming these are uniquely determined by In(Mi) and by

U(Mi). Given the initial condition and colours for the control registers, the computation of Mis its steady state, that is the set of colours in In(MN ) where N ≥ 0 is such that each initialregister has the same colour as its corresponding terminal register in Mn for all n > N . A steadystate can be diagrammatically described via a colouring of the closure of M (concatenating eachterminal vertex with its corresponding initial vertex). Conversely, M may compute the set ofinitial conditions for which a steady state exists.

Special cases of the above computational paradigm have been studied in (Kauffman, 1994,1995). His tangles have a single open process, and these represent feedback loops which are aresearch interest of Kauffman and a primary ingredient in cybernetic sciences. Using a quandle


u0 u0u1 u1u2 u2u3 u3u4 u4un un

x0:n x0:n

x1:n x1:n

x2:n x2:n

Figure 6. Nested machines

colouring, Kauffman showed that such long knots underlie a class of automata which can emulatemulti-valued logic and modular arithmetic computations. An example he considers is based oniterating a ‘trefoil machine’ M in which initial registers x0 and y0 are coloured a and b in somequandle Q whose underlying set underlies a field F and whose operation is a ⊲ b = 2b− a. Therecursion machine M attains a steady state if and only if 3(a − b) = 0, i.e. if and only if a− bis an element of order 3 in F .

Figure 6 shows some examples. The machines on the upper row are studied in Section 5.1.1,while the remainder of the section considers a machine which models a Markov chain.

Remark 5.1. Our theory does not account for machines with infinitely many interactions, sowe may assume that the nesting is large but finite. This assumption has nothing to do withwhether or not the recursion halts, whatever halting means in our context.

5.1.1. Basic straight line recursion. We investigate the computation of the equivalent machinesin the upper row of Figure 6. We colour these machines by the straight-line quandle over R,that is

(45) x ⊲ ydef= (1− s)x+ sy, ∀x, y ∈ R, for some s 6= 1.

The initial colours in both machines are given as u0, u1, . . . ∈ R. The terminal colours are thencomputed to be:

(46) x0:n = (1− s)nu0 + sn−1∑

i=0

(1− s)iun−i

which can be expressed concisely as x0:i = x0:i−1 ⊲ ui with x0:0 = u0. This recursion describesa dynamical system, or more precisely, an equivalence class of such systems whose behavior isdictated by the fixed quandle parameter s and by the inputs ui, i = 0, 1, 2, . . ..

Equation 46 expresses x0:n as a sum of an effect of the initial condition u0 with a discrete-timeconvolution of (1 − s)i with the inputs ui, where i is the discrete-time index. This expressionmay be viewed as a generating function encoding information about the inputs, by rewriting it


as:

(47) x0:n =

n∑

i=0

wi(s)un−i.

The coefficients wi(s), i = 0, 1, 2, . . . all are machine invariants, i.e. for equivalent machines theyare the same.

Aside from x0:n, the machines also compute x1:n, . . . , xn:n. These all are the outputs of relateddynamical systems with increasingly smaller evolution histories. Thus, xk:n is the output of asystem whose initial state is xk:k = uk and whose evolution time-span is n− k, i.e. so far it hasprocessed n− k inputs.

5.1.2. Markovian links. We next present a more involved example of a recursive computation.Our machine will be coloured by a quandle (Q,B) where Q = R and B is the set of binary

operations:

(48) x ⊲s ydef= (1− s)x+ sy, x, y ∈ R,

for 1 6= s ∈ R.Note that (Q,B) has no non-trivial automorphisms.

Remark 5.2. Our (Q,B) satisfies the defining relation of a distributive Γ–idempotent right quasi-group (Buliga, 2009, 2011a). Namely:

(49) x ⊲s (x ⊲t y) = x ⊲st y ∀x, y ∈ Q ∀ s, t ∈ Γ.

It is not, however, a distributive Γ–idempotent right quasigroup, because elements of B areindexed by elements of R− {1}, which is not a commutative group under multiplication.

v1i

v2i v1i+1

v2i+1

⊲1

⊲2

sub-machine Mi

Mi−1 Mi Mi+1

π1π2

v2i−1

v1i−1

v1i+2

v2i+2

coloured closure (stationarity)

recursion M

Figure 7. A recursive Q–coloured machine and its steady-state.

Consider the Q–coloured recursion machine M built out of concatenating identical copies M0,M1, M2, . . . of the machine pictured in the upper left corner in Figure 7 by concatenating thetwo terminal registers v1i+1 and v2i+1 in Mi to their namesake initial registers in Mi+1. Eachmachine Mi involves two operations ⊲s1 and ⊲s2 with s1, s2 6= 1. We abbreviate the namesof these operations to ⊲1 and ⊲2 correspondingly. The operation represented by each agent iswritten beside it.

For the specified concatenation to be defined, the following relation between Out(Mi) andIn(Mi) must be satisfied:

(50){v1i+1 = v1i ⊲2 v

2i , v

2i+1 = v2i ⊲1 v

1i

}−→ vi+1 =

[1− s2 s2s1 1− s1

]

︸︷︷︸

P

vi,


where videf=

[v1iv2i

]

.

To avoid degenerate cases, we assume henceforth that s1, s2 6= 0.All entries of the one-step transition matrix P are non-negative, and each of its rows sums to

1. A matrix with such properties is said to be (right) stochastic.The Perron–Frobenius Theorem for stochastic matrices tells us that P a unique largest eigen-

value equal to 1 whose corresponding eigenvector π has strictly positive entries. Again by

Perron–Frobenius, for any vector v0 of probabilities satisfying∑

j vj0 = 1, the homogenous irre-

ducible Markov chain with one-step transition matrix P converges to π irrespective of the initialdistribution v0:

(51) limi→∞

P iv0 = π.

The recursion machine M represents an homogeneous irreducible Markov chain whose one-step transition matrix P is given by (50). We have shown that M has a steady-state, which wemay describe by ‘closing’ a machine Mi:

(52) In(Mi) = Out(Mi) −→ π = Pπ

Thus, π =

[π1

π2

]

is the eigenvector of P corresponding to the eigenvalue 1.

Remark 5.3. In the special case s1 = s2, matrix P is doubly stochastic.

5.1.3. Feed-forward. Figure 7 gives machine analogous to a homogeneous irreducible Markovchain, for which a steady-state colouring is always attained. Such machines are said to be(externally) stable. The machine M is also internally stable, meaning for any concatenation of

machines that gives rise to M , each transition matrix describing a concatenation is stochastic.In this section and the next, we shall exhibit equivalent machines to M which are not internally

stable. To the best of our knowledge, there is no competing formalism in the literature for whichto discuss equivalent Markov chains which may or may not be internally stable.

Consider a machine M ′ ∼ M built from concatenating (‘stacking’) copies M ′0,M

′1, . . . of the

upper machine M ′ in Figure 8 by concatenating each register in M ′i with its namesake register in

M ′i+1. The ‘feed-forward machine’ M ′ is created by sliding the concatenated output strand v2i+1

of Mi all the way across the outputs of Mi+1, crossing over the inputs of Mi. This overcrossing

strand, pictured as a thickened line, acts as an agent via ⊲3def= ⊲s3 , where s3 6= 1 is some real

number. Metaphorically, we are using a colour v2i+1 ‘from the past’ to manipulate colours v1i+2

and v2i+2 ‘in the future’.By the equivalence of M with M ′, we know that:

(53) vi+2 = P 2vi

where, as before, vi =

[v1iv2i

]

. Unlike in M , the colours vi+1 and of Pvi need not coincide in

M ′. Writing Pi for the matrix such that vi+1 = Pivi, we now obtain, instead of the relationvi+1 = Pvi for M , the pair of relations:

(54) v2i = P1v2i−1, v2i+1 = P0v2i, where P0P1 = P 2.

Thus, the one step transition matrices in M all equal P , while in M ′ the transition matrixfrom vn to vn+1 is Pn mod 2.

We compute P0 and P1 explicitly:

(55a) P0 =

[(1− s2 − s1s3)(1− s3)

−1 (s2 − s3 + s1s3)(1− s3)−1

s1 1− s1

]

(55b) P1 =

[(1− s2)(1− s3) s2(1− s3) + s3

s1(1− s3) (1− s1)(1− s3) + s3

]

.


Mi

Mi

Mi+1

Mi+1

v2i

v2i

v1i

v1i

v2i+1

v2i+1

v1i+1

v1i+1

v2i+2

v2i+2

v2i+2

v1i+2

v1i+2

feed-forward machine M ′

feed-back machine M ′′

Figure 8. Feed-forward and feed-back equivalent machines.

The important point is that P0 and P1 may no longer be stochastic as some of their entriesmay be negative or greater in magnitude than 1. But P0P1 = P 2 is stochastic and well-behaved.So perhaps the computation of internal colours in each M ′

j should be thought of as abstract. Inparticular, the internal colours in vi+1 are not bounded as s3 gets closer to 1, and may thereforenot represent probabilities. Thus, M ′ and M ′ are internally unstable.

5.1.4. Feed-back. Next consider the feed-back machine M ′′ in Figure 8. It is formed by slidingthe output strand v2i+2 all the way back across the inputs of Mi. It is as though a ‘future’ registermanipulates ‘past’ ones. Similarly to the feed-forward machine, the feed-back machine may beinternally unstable. Its structure is yet more intricate in that it resembles a regulating controlloop such as those which are encountered in the theory of dynamical systems and in cybernetics.

We again compute the relations between Out(M ′′) and In(M ′′) required for concatenation.For the feed-back machine also v2i = P1v2i−1 with:

(56) P1 =

[(1− s2)(1− s3) s2(1− s3) + s3

s1(1− s3) (1− s1)(1− s3) + s3

]

.

For the transition from v2i to v2i+1, we compute:

(57) v2i+1 =

[(1− s2)(1 − s3)

−1 s2(1− s3)−1

s1(1− s3)−1 (1− s1)(1 − s3)

−1

]

︸︷︷︸

P ′′

0

vi +

[0 −s3(1− s3)

−1

0 −s3(1− s3)−1

]

︸︷︷︸

T

v2i

We deduce that:

(58) v2i+1 =(P ′′0 + TP 2

)v2i, v2i+2 = P1

(P ′′0 + TP 2

)v2i

Moreover, (53) attests that P 2 equals P1

(P ′′0 + TP 2

), because both take vi to vi+2. Hence

we find that:P 2 = (I − P1T )

−1 P1P′′0

which leads to

(59) v2i+1 = (I − P1T )−1 P1P

′′0 v2i


Relations of the form (59) are encountered in the theory of dynamical systems, where theymanifest a regulating procedure known as a closed (control) loop. In the context of machines,a closed loop is interpreted as follows. Any machine equivalent to the internally stable machineM is stable, but not necessarily internally stable. We might imagine islands of instability inan externally stable cosmos (machine). Feed-back and feed-forward machines which are notinternally stable regulate their behavior so as to become externally stable. In our example,the one-step transition matrices were not stochastic, but the two-step transition matrices arestochastic. Following the common practice in control theory, a figurative description underlyingthe feed-back machine is suggested by Figure 9.

Pi+1P ′′i

(P ′′i )

−1Ti

+

vi

vivi+1

vi+2

vi+2

P 2

feed-back machine M ′′ two concatenated machines M

Figure 9. Closed loop representation of the feed-back machine and its equivalent(open loop) counterpart.

5.2. Adiabatic quantum machines. Some paradigms for quantum computation do away withthe conventional circuit model. Adiabatic quantum computation is one such approach (Farhiet al., 2000). The idea behind it rests on the Adiabatic Theorem in Quantum Mechanics whichroughly states that a (quantum) system remains in its ground state when subjected to environ-mental perturbations, as long as these act slowly enough and as long as there is a gap betweenthe ground state and the rest of the Hamiltonian’s spectrum. Adiabatic quantum computationmakes use of this fact by adiabatically evolving a simple Hamiltonian H0, which can be thoughtof as a problem whose solution (the ground state) is easy, into a different and perhaps morecomplicated Hamiltonian H1 whose ground state is the solution to the problem at hand. Thecomputation works by initializing the system in its ground state, the ground state of H0, andslowly evolving its Hamiltonian to H1. This process is called quantum annealing. By the adi-abatic theorem, the system remains in its ground state throughout the evolution process, andthe computation concludes at the ground state of H1, that is the sought-after solution.

The computational difficulty of this procedure is proportional to the minimal energy gap

between the ground state and the rest of the spectrum, namely to gdef= λ1−λ0, where λi+1 ≥ λi

are the underlying energy eigenvalues of the Hamiltonian.We introduce an adiabatic quantummachine (AQC). Strictly speaking, this is a one-parameter

family of tangle machines. For s ∈ [0, 1), consider the quandle Qs whose elements are self-adjoint

operators over a Hilbert space of dimension 2N , and whose operation is x ⊲ ydef= (1 − s)x+ sy.

In most cases N stands for a number of qubits, and N is always fixed. As s evolves from 0 to1, a machine M0 coloured by a trivial quandle Q0 evolves through machines Mt coloured by Qt.The limit s → 1 no longer gives a quandle coloured machine, but the machine is designed sothat the colours in the terminal registers of M1 represent the solution to the computation.

Remark 5.4. A similar paradigm appears in the context of biological computation in work ofKauffman and Buliga (Buliga, 2013; Buliga & Kauffman, 2013). This follows from (Kauffman,1995), in which Kauffman argues for knot and knotted graphs as fundamental logical objects innonstandard set theory and in lambda calculus.

5.2.1. Single interaction adiabatic quantum machine. The standard notion of adiabatic compu-tation corresponds to a machine with a single interaction, as in Figure 10. A general AQC hasmultiple interactions, which we should consider as adiabatic computers working in conjunction


H0

H1

Hout

Figure 10. An AQC with a single crossing.

to arrive at a solution. We will not give many details about adiabatic quantum machines inthis paper— we will only demonstrate what we have set out to: the way in which machineequivalence makes a difference in terms of computation. Our example involves only a singlequbit.

Let σx =

(0 11 0

)

and σz =

(1 00 −1

)

denote two out of three Pauli matrices, expressed with

respect to the basis of C2 consisting of the eigenvectors of σz. We use the standard notation, inwhich subscripts denote spin axes. Let 1 denote the identity operator. Our adiabatic computeris designed to output the ground state |1〉z. Choose the terminal Hamiltonian to be:

(60) H1def=

1+ σz2

=

(1 00 0

)

= |1〉〈1| .

Choose the initial Hamiltonian to be:

(61) H0def=

1− σz2

=

(0 00 1

)

= |0〉〈0| .

The ground states of H0 and of H1 are, respectively, |0〉z and |1〉z.At time s, our crossing has input H0, agent H1, and output

(62) Hout(s) = H0 ⊲ H1 =1+ (2s − 1)σz

2=

(s 00 1− s

)

.

Starting with Hout(0) = H0, the system evolves Hout(s) towards H1 as s approaches 1. But thecomputation turns out to be infeasible because the minimal energy gap along the evolution pathvanishes, g (Hout(1/2)) = 0. This is due to the problem Hamiltonians H0 and H1 sharing thesame eigenbasis, causing the energy levels to cross one another. To stress this fact we say thatsuch a machine is (computationally) infeasible.

5.2.2. Multiple interaction adiabatic quantum machine. The level crossing problem described inSection 5.2.1 can be avoided by extending the machine to include more than one interaction.Equivalent variants of the proposed AQC machine are given in Figure 11. All machines have

registers coloured H0, Hout, and H 1

2

. These colours do not depend on s. Set H 1

2

def= σx. The

terminal colour Hout has the following form:

H 1

2

H 1

2

H 1

2H0 H0H0

H1

H1

H1

H 1

2

⊲ H1 H 1

2

⊲ H1H 1

2

⊲ H1

H ′′H ′

H ′

Hout HoutHout

G

abstract feasible infeasible

Figure 11. Equivalent adiabatic quantum machines.


(63) Hout = (H0 ⊲ σx) ⊲ H1 = (1− s)2H0 + s(1− s)σx + sH1 =

[s s(1− s)

s(1− s) (1− s)2

]

Thus if we write Hout(s) for Hout at time s, then Hout(0) = H while Hout(s) → |1〉〈1| = H1

for s→ 1. The ground state of H1 is the sought-after solution.The simple calculation of the classical adiabatic computer H0 ⊲ H1 in Section 5.2.1 has been

replaced by the more involved computation of Equation 63. We find that

(64) g (Hout(s)) =[(s+ (1− s))2 − 4s(1− s)3

] 1

2

and mins g (Hout(s)) >25 . Thus we have solved the level crossing problem, and the final com-

putations of each of the machines in Figure 11 are feasible.A general AQC machine is fundamentally different from the single-crossing ‘classical adia-

batic computation’ in that it has intermediate stages at which intermediate Hamiltonians arepresent, describing neither the initial nor the terminal problem. This may have practical implica-tions. The AQC machine paradigm is suitable for describing a network of adiabatic computers,or equivalently a multi-core adiabatic processor containing a number of interacting quantum(sub)systems which together undergo adiabatic evolution. In such a setting, the concept ofmachine equivalence may allow us to maximize the minimal energy gap during the computationbetween the ground state and the rest of the spectrum.

Remark 5.5. We will prove elsewhere that for every initial Hamiltonian H0 there exists a ‘controlHamiltonian’ H 1

2

such that the energy gap of the feasible AQC machine in Figure 11 is greater

than or equal to the energy gap of the single-crossing AQC, and that this inequality is strict forH0 6= H1.

As we are no longer interested only in the output Hout but also in the system as a whole,the Adiabatic Theorem should be applied also to all of the intermediate Hamiltonians in theAQC machine. With this in mind, let us examine the behavior of the equivalent machines fromFigure 11.

The middle machine has two intermediate Hamiltonians that depend on s, namely, σx ⊲ H1

and H ′ = H0 ⊲ σx, written explicitly as:

(65) σx ⊲ H1(s) =

[s (1− s)

(1− s) 0

]

, H ′(s) =

[0 ss (1− s)

]

Thus, g (σx ⊲ H1(s)) =[s2 + 4(1 − s)2

] 1

2 and g (H ′(s)) =[(1− s)2 + 4s2

] 1

2 , both which have

minimum energy gap mins g ≥ 2√5. As the energy gaps across the middle machine g(H ′) ,

g(σx ⊲ H1), g(Hout) are all non-vanishing throughout the adiabatic evolution, we conclude thatthe central machine in its entirety represents a feasible computation.

Conversely, the machine on the right possesses no advantage compared to the classical adia-batic scheme. One of its Hamiltonians, H ′′ = H0 ⊲H1, has a vanishing energy gap for s = 1

2 . Sotaken as a whole, the machine on the right represents an infeasible computation.

The machine on the left in Figure 11 presents a computation which may be considered un-physical. Calculating the energy levels of the Hamiltonian G = σx ⊳ H0, where ⊳ stands for theinverse of the quandle operation, gives

(66) G(s) = (1− s)−1 (σx − sH0) =

[0 (1− s)−1

(1− s)−1 −s(1− s)−1

]

.

This matrix has at least one negative eigenvalue for any s ∈ [0, 1). Thinking of eigenvalues asenergy levels in a system, such a result seems absurd. Nonetheless, the machine solves exactlythe same problem as its feasible counterpart.

5.3. Machines and information. The concept of computation is broad, and extends beyondcalculating the answer to a prescribed problem. Perhaps the most general characterizationof computation is that it is ‘a manipulation or processing of information’. Computation andinformation are intertwined, and these two concepts rely heavily on one another.


In this section, machines are conceived of as a class of networks for distributed informationprocessing. Colours represent information entropies. The information processing capacity asso-ciated with an interaction, called its local capacity, is defined to be the mutual information ofthe initial and terminal colours. A machine M represents a network within which informationis processed and sent further down to other interactions or registers. A machine equivalent toM is a network which, as a whole, exhibits an information processing capacity the same as M ,but whose local capacities may be different.

Our definitions in this section follow (Cover & Joy, 2006). We take note that the reader maynot be fully acquainted with this field and hence maintain expositions as informal as possible.

We colour machines by the quandle (Q,B) of Section 5.1.2, whose elements are real numbers,with an operation

x ⊲s ydef= (1− s)x+ sy

for each s 6= 1. To recap, each register r is coloured by a real number nr 6= 1, and it acts oneach of its patients via either ⊲nr

or ⊳ nr. In this section, elements of Q represent entropies.

5.3.1. Preliminary definitions. An information channel is an apparatus through which messagesare transmitted from one location to another. In practical situations, a message entering thechannel on one end will emerge corrupted on the other end. It is convenient to think of a messageas a sequence of zeroes and ones. An information channel is characterized by its capacity, that isthe maximal rate at which messages may be transmitted with a ‘negligible’ loss of information.Entropy is conceived of a measure of information, or rather, of uncertainty. If a message isconstructed by sampling N independent identically distributed (iid) binary random variables,then Shannon’s Source Coding Theorem (Shannon, 1956) tells us that, for typical sequences,the entropy times N is nearly the number of information units (e.g. bits) required to encode amessage so that it can reliably be recovered by a receiver.

Compressible messages exhibit some kind of pattern (H < 1), and these admit shorter de-scriptions than the length of the message itself. This is the key principle underlying messagecompression. Incompressible messages are messages for which randomness inhibits descriptionsshorter than the message’s own length (i.e. H ≥ 1).

A general computing device (e.g. a universal Turing machine) requires two distinct inputs.The first input X0 is a stream of data that is read and manipulated by the machine accordingto instructions given by the second input X1. Both inputs X0 and X1 and the result of acomputation Xout are all assumed to be typical binary sequences.

5.3.2. Information processing by machines. A machine describing an information processingnetwork is a concatenation of interactions. Each of its registers is coloured by a real numberrepresenting an entropy. The colour of an agent register represents the entropy of a programmetypical sequence, while colours of input registers represent entropies of data typical sequences.The agent register is equipped with a parameter s ∈ (0, 1), which may represent some (input-independent) property of the computing device itself. The colour of the output correspondingto input H(X0) is:

(67) H(X0) ⊲s H(X1)def= (1− s)H(X0) + sH(X1).

If H(X0) > H(X1) then the output entropy is strictly lower than the input entropy, i.e.H(X0) ⊲s H(X1) < H(X0).

Thus, the computing device computed Xout by applying the instruction data steam X1 to theinput data stream X0, and the entropy of Xout is H(X0) ⊲s H(X1). See Figure 12.

5.3.3. Capacity. In this section we describe various capacities associated to machines, whichprovide a measure of how ‘good’ a computation is.

Our analysis of a computing device whose internal workings are unknown to us focusses ondiscrepancies between its input and output streams. Suppose that we wish to know if the com-putation is meaningful in some sense. If no additional restrictions are made, then “meaningful”could in essence mean that computations produce intelligible answers which could read off by ahuman operator. Translating this requirement into the language of preceding paragraphs, the


PSfrag

H(X0)

H(X1)

H(X0)⊲sH(X1)X0

X1

Xouts

Figure 12. The computation, and the corresponding interaction between entropies.

output stream is expected to appear ‘less random’ than the input stream. According to thisparadigm, computation and compression are literally the same thing. A ‘good computation’ isone which compresses X0 as much as possible, given X1. In the language of information theory,the optimal output Xout has entropy equal to the conditional entropy H(X0 | X1). The channelcapacity of the computing device is defines as the mutual information:

(68) I(X0 : X1)def= H(X0)−H(X0 | X1).

The local capacity of a process is the entropy of its initial register minus the entropy of itsterminal register. For example, for a crossing with a single input-output pair:

(69) Caps

In Out

def= H(X0)

︸︷︷︸

In

−H(X0) ⊲s H(X1)︸︷︷︸

Out

The global capacity of a machine is the set of all capacities of its processes.The computation represented by a process is optimal if the capacity of the process is equal

to the mutual information:

(70) H(Xin)−H(Xout) = I(Xin,Xout).

This occurs when H(Xout) = H(Xin | Xout).

5.3.4. Equivalent machines. Consider the three equivalent machines in Figure 13.

H(0) H(0)H(0)H(1) H(1)H(1)

H(2)

H(2)

H(2)

H(0)⊲tH(2) H(0)⊲tH(2)H(0)⊲tH(2)

H(1|2)H(1⊲0)

H(1⊲0)

H(1|0,2) H(1|0,2)H(1|0,2)

⊲s

⊲s

⊲s

⊲t

⊲t

⊲t

abstract locally suboptimal locally optimal

Figure 13. Equivalent machines with the same global information processingcapacities. The middle and right machines are feasible whereas the left machineis abstract. While all of them are globally optimal only the rightmost machineis also locally optimal.

As the three machines are equivalent, they have the same global capacities. But the capacitiesof their interactions are quite different, and the leftmost machine represents an impossible,abstract computation.


Set the following values of t and s:

(71) t =H(1)−H(1 | 2)H(1)−H(2)

, s =H(1 | 2)−H(1 | 0, 2)

H(1 | 2)−H(0) ⊲t H(2)

In order to assure that t, s ∈ (0, 1), we choose our entropies so that:

(72) H(1 | 2) > H(2), H(1 | 0, 2) > H(0) ⊲t H(2)

which essentially describe the extent to which the sources, X0, X1, and X2, are statisticallydependent. This is illustrated by the following Venn diagrams:

(73)

H(1 | 2) H(1 | 0, 2)

X2X2

X1X1

X0X0

All three machines are globally optimal, but the local capacities for the three machines inFigure 13 are different. In the rightmost machine, by our choices of t and s, each interactionis itself locally optimal— see Figure 14. This is no longer true for the middle machine, which

has a register labeled H(1 ⊲ 0)def= H(1) ⊲s H(0), which may not equal H(1 | 0). In this case,

the middle machine contains a non-optimal interaction. The left machine involves the inverseoperation ⊳ s, so that its colour H(1) ⊳s H(0) might be negative. The idea of negative entropiesmay sound absurd, but nevertheless the leftmost machine in Figure 13 is equivalent to a machineall of whose computations are feasible, and in fact even optimal. In view of this, we may thinkof this machine as a sort of abstract information processing scheme.

H(0) H(1)

H(2)X1

X1

X1

X2

X2

X2

X0

X0

X0H(0)⊲tH(2)

process capacity in I(M)

H(1|0,2)

local capacities at crossings Cap

I(1 : 2)

I(1 : 2 | 0)

I(1 : 2, 0)

Figure 14. Optimal information processing along a process P1 in the rightmost(locally optimal) machine in Figure 13.

Remark 5.6. Define the transition operator of a Q–coloured machine M to be the n× n matrixT (M) taking the vector of initial colours to the vector of terminal colours. It seems to bethe case that topological invariants of M are related to algebraic invariants of T (M), whichsuggests one way in which the formalism of machines might be used to understand real-worldinformation processing networks. For a small result along these lines, proved elsewhere, we makethe following definition. A useless machine is one whose global capacity is a set of zeroes. Thusthe colour of the initial register of each open process in a useless machine equals the colour


of its terminal register. The Shannon capacity of a machine is defined in the sequel to thepresent paper. The theorem is that a useless machine whose transition operator is irreduciblehas Shannon capacity 1 or

√2 (the smallest non-trivial Shannon capacity, i.e. 1

2 -bit).

6. Conclusion

In this paper, we have introduced tangle machines as a diagrammatic algebra uniting ideasin low-dimensional topology, causality, information, and computation. There is a natural localnotion of tangle machine equivalence. We have exhibited ways in which machine equivalencemay represent networks with identical global properties, but with different local properties,within a number of different paradigms of computation. Our vision is to model these and othercomplex real-world phenomena by machines, then to use machine equivalence to select a ‘best’machine (whatever ‘best’ means in that context), and then to perform a computation for that‘best’ machine which might not have been tractable for the machine that we started with.

Future work will discuss topological invariants of machines, will expand on our examples,will discuss statistical detection of machines inside data, and will discuss algorithmic aspects offinding a ‘best’ machine inside an equivalence class.


Figure 15. The unknot and the left-hand trefoil.

Appendix A. Knot theory background

This appendix provides the knot theoretical conceptual framework for the definition of atangle machine in Section 3, background for the topological constructions of Section 4 and ofthe sequel, and for the invariants defined in the sequel. Specifically, in Section 4, a machineplays the role of a ‘Gauß diagram’ for their ‘Reidemeister diagram’, which will be revealed inthe sequel in turn to be a projection of a network of jointly embedded spheres and intervals inR4.

The reader who is familiar with knot theory will find nothing new in this appendix beyonda small non-standard finesse in Section A.7, in which we define the connect sum operationcombinatorially, without cutting.

A.1. Knots and knot diagrams. The material in this section is standard and was known bythe late 1920’s. A reference is e.g. (Kauffman, 2001).

A knot (or an unframed knot) is a smooth embedding K : S1 → R3 of a circle in 3–space,where S1 is parameterized as S1 ≃ R/2πZ. This means that a knot has a basepoint (the cosetof 0), is oriented (by the orientation induced from R), and that we can meaningfully specify apoint on a knot. Abusing notation, K(S1) is traditionally shortened to K, so that K mightmean either an embedding or its image, depending on the context. To simplify figures, we drawbasepoints and orientations only when they make a difference.

Example A.1. The knot O def= (cos t, sin t, 0) is called the unknot. See Figure 15. It bounds an

embedded disc in R3, that is the unit disc on the xy plane.

Example A.2. The knotKdef= (sin t+ 2 sin 2t, cos t− 2 cos t,− sin 3t) is called the left-hand trefoil.

Roughly speaking, this is the knot obtained by forming a loop in a piece of string, passing a freeend up through the loop, and fusing the two free ends so that the knot cannot come loose. SeeFigure 15.

It is usually easier to study knot diagrams in a plane rather than to study knots in R3 directly.Choose a vector v, and denote its orthogonal plane P . If K intersects P then translate K in thedirection of v until K∩P = ∅. A Reidemeister diagram of K is projection of K on P whose onlysingularities are regular double-points, together with additional information at crossings calledcrossing information which tells us which of the two preimages in R3 of a double-point on theknot diagram lies closer to P . A Reidemeister diagram is usually simply called a diagram of K.The non-standard terminology ‘Reidemeister diagram’ is to stress that we shall later considerequivalence classes of these under ‘Reidemeister moves’.

Remark A.3. By Sard’s Theorem, the set of unit vectors v that give rise to planes P whose ‘knotdiagrams’ have triple-points, self-tangencies, or worse, is of measure zero on the unit sphere (seee.g. (Kosinsky, 2007)). Thus, loosely speaking, a randomly chosen vector v gives rise to a validknot diagram for K ‘with probability 1’.

Consider a small disc D around a double-point of a diagram D of K. Crossing informationseparates D ∩K (which looks like a small letter ‘x’) into three line segments: An overcrossingarc, and two undercrossing arcs. Our convention is that the preimage of the double point onthe overcrossing arc is closer to P than the preimage of the same double point on the arc whichwe break. Breaking the knot diagram at undercrossings separates it into arcs.


R2 R1

Figure 16. Ambient isotopy of a framed knot. The rightmost knot is notambient isotopic to the others.

arc 1

arc 2arc 3

A variation on the notion of a knot is a notion of a framed knot. A framed knot is a smooth

embeddingK : S1×I → R3, where Idef= [0, 1] denotes the unit interval. We think of K(S1×{0})

as being ‘the knot ’ and of K(S1 × {1}) as being ‘the framing curve’ (a framing is a continuouschoice of unit normal vector for each point in K, which is exactly the structure that the framingcurve K(S1 × {1}) induces on the knot K(S1 × {0})).

Two knotsK1 andK2 are ambient isotopic if there exists a smooth map h : R3×I → R3 whoserestriction ht : R3×{t} → R3 is a homeomorphism for all t ∈ I, such that h(K1×{0}) = K1 andh(K1×{t}) is a knot for all t ∈ [0, 1], and h(K1×{1}) = K2. Ambient isotopy of a framed knothas a parallel definition, and is illustrated in Figure 16. Ambient isotopic knots are consideredequivalent.

A standard sloppiness in knot theory is to consider equivalent knots to be the same, so thatK may denote not merely a single knot, but an equivalence class of knots. We confuse freelybetween knots and knot equivalence classes in all that follows.

The fundamental fact about knot equivalence is the Reidemeister Theorem.

Theorem A.4 (Reidemeister Theorem). Two framed (unframed) knots K1 and K2 are equiva-lent if and only if we can pass from any diagram D1 of K1 to any diagram D2 of K2 by a finite


R2

R3

R1

Figure 17. Reidemeister moves for knots, links, and tangles. Orientations arearbitrary. To execute a Reidemeister move, cut out a disc inside a knot diagramcontaining one of the patterns above, and replace it with another disc containingthe pattern on the other side of the Reidemeister move. This is made precise inRemark A.12.

sequence of Reidemeister II and III (and I) moves, which for short are written as R2 and R3(and R1). These are local modifications of the knot diagrams which are supported inside a smalldisc in P , illustrated in Figure 17.

A.2. Coloured knots. As graphs can be studied by defining flows on them, so knots can bestudied by colouring them. It is traditional to think of a colouring of a knot as extra structure,used as an anchor to probe the knot’s properties. In this paper we have taken the opposite pointof view. For us, a knot was a container for its colours, and the interactions which it specifiedfor its colours provided a basis for our model of exchange of information between interactingprocesses.

The ‘palette’ with which a (framed) knot can be coloured is called a (rack) quandle, asdefined in Definition 3.2. A Q–coloured (framed) knot is a knot diagram whose arcs are labeledby elements of a quandle (rack) Q, subject to the crossing rule:

x x

y y

x ⊲ yx ⊳ y

Figure 18 is an example of a coloured knot.The axioms of a quandle guarantee that a colouring is preserved under Reidemeister moves,

and therefore that we can colour ‘equivalence classes of knots’:


0

0

1

1

1

2

2

Figure 18. A knot coloured by a quandle has 3 elements {0, 1, 2} (‘red’, ‘blue’and ‘green’) with operation a ⊲ b

def= (2b− a) mod 3.

Reidemeister I:

xx

x x ⊲ x

In the left diagram, there is one arc and there are no crossings, therefore only onecolour appears. Therefore the right diagram may only contain one colour, forcing x ⊲ xto be equal to x. This equality indeed holds if Q is a quandle, because all colours of Qare idempotent.

Reidemeister II:

x

yx

y x

x

y

y

x ⊲ y

In the left diagram of this figure, the right strand does not cross under anything, andtherefore it has only one colour. Therefore the top colour on the right strand in theright diagram (the colour designated by a bold letter) must also be equal to y. So ifa ⊲ x is equal to y ⊲ x, then R2 tells us that a must be equal to y. Indeed, one of theaxioms of a rack is that the ⊲y operation induces an automorphism of Q as a set. Inparticular, the operation ⊲ has an inverse ⊳ .

Reidemeister III:

(x ⊲ z) ⊲ (y ⊲ z)

y ⊲ zy ⊲ z

zz

zz x

yy

x ⊲ zx ⊲ y

x

(x ⊲ y) ⊲ z

The top colours on the bottom-most strand must be equal, which tells us that

(a ⊲ b) ⊲ c = (a ⊲ c) ⊲ (b ⊲ c) for all a, b, c ∈ Q.

In other words, R3 tells us that the automorphism of sets ⊲c : Q→ Q distributes withrespect to the ⊲ operation, and is thus in fact an automorphism of (quandles) racks forall c ∈ Q. This is indeed an axiom of a rack.

There are different orientations, and mirror images, possible for Reidemeister moves, but theverification that quandle colourings respect these moves is the same.


Two coloured knots are considered equivalent if they differ by a finite sequence of Reidemeistermoves, together with a composition of ρ with an automorphism α : Q→ Q of the rack Q.

The fundamental quandle (rack) Q(K) of a (framed) knot K is obtained by taking one gener-ator for each arc of a knot diagram D of K, with relations given by the crossing rule and by theaxioms of a quandle (rack). Any Q–colouring of K for any quandle or rack Q can be obtainedby composing the Q(K)–colouring of K with a quandle (rack) homomorphism from Q(K) to Q.This means that a coloured knot is a knot for which we ‘focus in’ on one part of the informationcontained in Q(K). See Figure 19.

x

yx ⊲ y

Figure 19. The fundamental quandle of the trefoil has two generators x and y,subject to the relations (x ⊲ y) ⊲ x = y and y ⊲ (x ⊲ y) = x.

Remark A.5. Because two knots are ambient isotopic if and only if any two of their diagramsdiffer by Reidemeister moves, which happens if and only if they share the same fundamentalquandle or rack, it follows that any local move on a knot which preserves any colouring of theknot (equivalently, which preserves the fundamental quandle/rack) is a finite combination ofReidemeister moves.

A.3. Links. A generalization of a knot is a link, that is a smooth embedding in R3 of a finitecollection of circles. An example of a link is given in Figure 20. A knot is a link with a singlecomponent. All theorems and constructions of the previous sections generalize to links mutatismutandis.

Figure 20. The Borromean Rings link, whose linking matrix is the 3× 3 zero matrix.

The linking matrix of a ν–component link L in S3 is the symmetric ν × ν matrix Link(L)whose ijth entry is the number of times component i passes right to left under component j

i.e.j

I

minus the number of times component i passes left to right under component j i.e.

j

I

. Entries of linking matrices are called linking numbers. Matrix Link(L) is a topological

invariant of a framed link L (i.e. no R1 moves), and the matrix Link0(L) obtained from Link(L)by setting all diagonal entries (framings) to 0 is a topological invariant of an unframed link L(i.e. with R1).

A link L is said to be split if there exist two disjoint embedded balls B1 and B2 each containingat least one component of L, which together contain all components of L.


Example A.6. The link

is split and has linking matrix

(0 00 2

)

.

Example A.7. The link

is non-split and has linking matrix

(0 −1−1 0

)

.

A.4. Gauß Diagrams. A Gauß diagram of a knot is a directed circle with basepoint (the skele-ton), together with a collection of directed chords. To obtain a Gauß diagram for a Reidemeisterdiagram D of a knot K, start from the basepoint, and walk along the knot K in R3 in the di-

rection of its orientation. When you reach a point a1def= K(e

2π i

θ ) which projects to a crossingin D, mark the corresponding point on the skeleton S by a+1 if a projects to the overcrossingarc, and a−1 if a1 projects to the ‘meeting point of the undercrossing arcs’. Do the same thingfor the next preimage of a double point, and so on, until you arrive back at the basepoint of

K. If D had k crossings, we now have 2k points(

(a+1 , a−1 ), (a

+2 , a

−2 ), . . . , (a

+k , a

−k )

)

marked on

S. Finally, draw a chord from a−i to a+i for each i = 1, 2, . . . , k.For coloured knots, while we are walking along the knot, assign to the corresponding point in

S the colour as the strand of D which is the projection of the point at which we are standing.Assign to each directed chord the colour of its tail. See Figure 21.

Figure 21. A coloured knot diagram and its Gauß diagram.

Reidemeister moves for Gauß diagrams are shown below (orientations of segments of theskeleton are arbitrary):


Figure 22. This Gauß diagram does not correspond to any knot. Instead itcorresponds to a virtual knot.

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R1

R2

R3

A.5. Virtual knots. Every knot has a Gauß diagram, but not all Gauß diagrams correspondto knots. See Figure 22.

Virtual knots are a generalization of knots which are in bijective correspondence with Gaußdiagrams whose skeleta are circles (Kauffman, 1999). As illustrated by Figure 22, virtual knotscan be drawn as knot diagrams in which some crossings are artifacts of the planar drawing, orare virtual. Virtual crossings ‘don’t really exist’, and virtual knot diagrams should be thoughtof as being analogous to planar drawings of non-planar graphs, which have crossings betweenedges which are merely artifacts of the planar drawing. Diagrams of virtual knots are consideredequivalent if they differ by Reidemeister moves, together with the following set of moves:

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VR1

VR2

VR3

SV

In the theory of virtual knots, these are called Virtual Reidemeister I, II, III, and the semivir-tual move correspondingly. Their effect of these moves is to change the position of the ‘virtual


Figure 23. A 3–coloured Granny Knot is a connect sum of two 3–coloured trefoils.

crossings’, reflecting the fact that virtual crossings are merely artifacts of planar drawings ofnon-planar objects.

There are framed and unframed versions of virtual knots— in the framed version, VR1 isallowed but R1 is disallowed— and coloured and uncoloured versions.

A.6. W–knots. W–knots are in bijective correspondence with Gauß diagrams whose skeletaare circles in which ‘tails commute’:

Satoh’s Conjecture states that w-knots are in bijective correspond with ribbon torus knots,which are a class of knotted tori in R4. This conjecture implies that two ribbon torus knots areambient isotopic if and only if their diagrams differ by a finite sequence of Reidemeister movesR1, R2, and R3, Virtual Reidemeister moves VR1, VR2, and VR3, the semivirtual move SV,plus an additional undercrossings commute move UC:

UC

W–knotted objects were defined in (Fenn, Rimanyi, & Rourke, 1997) and were studied furtherin (Satoh, 2000) and in (Bar-Natan & Dancso, 2013).

There are framed and unframed versions, and coloured and uncoloured versions of virtual andof w–knotted objects.

A.7. Connect sums of knots. A knot K is said to be a connect sum of K1 and K2 if thereexists a Gauß diagram D of K and a partition C = (C1, C2) of the set of chords C of D, such thatno chord from C1 intersects a chord from C2 in D, deleting C1 from D gives a Gauß diagram forK1, and deleting C2 from D gives a Gauß diagram for K2. Knots K1 and K2 are called factorsof K. See Figure 23 for an example.

Remark A.8. The above definition for a connect-sum of Gauß diagrams is equivalent to thestandard definition which splits D itself into two parts.

If any direct sum decomposition K = K1 K2 of a knot K satisfies that either K1 or K2 isthe unknot (equivalently, that either K1 or K2 has some Gauß diagrams without chords), thenK is said to be prime.

The topological definition of connect sum is that K = K1 K2 if there exists a sphere Sdividing S3 ≃ R3 ∪ {∞} into two balls B1 and B2, such that S ∩K consists of two points, andtaking the identifying the boundary of the closure of B1 to a single point gives K1 ⊂ S3, andidentifying the boundary of the closure of B2 to a single point gives K2 ⊂ S3.

Theorem A.9 (Unique prime decomposition). A knot K has a prime decomposition

(76) K = K1 K2 · · · Kk

which is unique up to permutation of factors.


Figure 24. Tangles.

Remark A.10. All known proofs of unique prime decomposition make use of the topology ofcodimension 2 embeddings, and are not combinatorial. Proving existence and uniqueness ofprime decompositions combinatorially is difficult because both the algebra of knot diagramsmodulo Reidemeister moves is difficult, and also the structure of racks and quandles is difficult.

A.8. Tangles. The word ‘algebra’ traditionally invoked the mental image of a strings of symbols.Knots are certainly not ‘algebraic’ in this sense, but perhaps they are algebraic in a wider sense,if we are allowed to concatenate symbols not only front-to-back, but also in an abstract way,like adding edges between vertices of a graph. A reference for this subsection is Section 4 of(Bar-Natan & Dancso, 2013).

See Figure 24 for examples of tangles. Define a tangle recursively.

• The single crossings and , and the null tangle , are tangles. Each of thesefundamental tangles may be thought of as a “lego brick”, which lives inside a regioncalled its support in the plane, whose boundary is called the firmament of the tangle.On the firmament, the tangle has studs (endpoints towards which an arc points) andholes (endpoints away from which an arc points).• A disjoint union of two tangles (placing one besides the other) is a tangle.• Concatenation in a tangle T is the operation of connecting a stud in T with a hole inT by an edge in the plane, and connecting their supports by regular neighbourhoods ofthe connecting edge. A concatenation of a tangle is a tangle. Depending on the knottedobject we are interested in defining, the may be restrictions on which concatenationsare allowable. For example, in the case of usual tangles, concatenation edges must benon-intersecting so that the resulting tangle diagram is guaranteed to be planar. SeeFigure 25. In the coloured case, we are only allowed to connect studs with holes of thesame colour. The null tangle can be concatenated with any circle component of thefirmament which has no studs or holes on it.

Thinking of concatenation as analogous to multiplication, a common philosophy in quantumtopology is to think of knotted objects as higher dimensional algebraic expressions, and to thinkof Reidemeister moves as higher dimensional algebraic relations on their generators (Kauffman,

1988; Jones, 1999). We have one extra move— to add and to take away the null tanglewithout studs or holes from the boundary of the tangle support gives a tangle. This adds ortakes away a circle boundary component from the firmament. An object obtained from a tangle

by a Reidemeister move or by attaching a null tangle is also a tangle, so for example isalso considered to be a tangle. Tangles are considered equivalent if they are related by a finitesequence of Reidemeister moves, and attachments or deletions of null-tangles.

Remark A.11. The set of tangles without studs and holes is in bijective correspondence withthe set of (usual/virtual/w-) links. The bijection is by erasing the firmament.

Remark A.12. Reidemeister moves can now be stated precisely. Remove null tangles and un-concatenate in order to cut a disc containing one side of a Reidemeister move out of the diagramD. The firmament of the resulting diagram has one more circle boundary component that the


Figure 25. Concatenating four different pairs of studs and holes on a tangle.

firmament of D, which contains studs and holes. Concatenate these with the studs and holes ofthe other side of the Reidemeister move, and fill in null tangles as required.

In this way, the space of knots, links, tangles, virtual tangles, w-tangles etc. are thought of ina universal algebra framework. In technical terms, each forms an algebra over a modular operad(Getzler & Kapranov, 1998).

Tangles also satisfy a Reidemeister Theorem, and their equivalence relation is also ambientisotopy. There are also framed and unframed, coloured and uncoloured versions.

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Division of Mathematics, School of Physical and Mathematical Sciences, Nanyang Technolog-

ical University, 21 Nanyang Link, Singapore 637371

E-mail address: [email protected]

Department of Mechanical Engineering, Ben-Gurion University of the Negev, Beer-Sheva 8410501,

Israel

E-mail address: [email protected]

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