Top Banner
4/10/08 Slide 1 Jo Ellis-Monaghan St. Michaels College, Colchester, VT USA e-mail: [email protected] website: http://academics.smcvt.edu/jellis-monaghan
26

St. Michaels College, Colchester, VT USA e-mail: jellis ...academics.smcvt.edu/jellis-monaghan/Talks/multivariable extension… · 1-partition 2-Partition 3-Partition. 4/10/08 Slide

Jun 13, 2020

Download

Documents

dariahiddleston
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Page 1: St. Michaels College, Colchester, VT USA e-mail: jellis ...academics.smcvt.edu/jellis-monaghan/Talks/multivariable extension… · 1-partition 2-Partition 3-Partition. 4/10/08 Slide

4/10/08 Slide 1

Jo Ellis-MonaghanSt. Michaels College, Colchester, VT USAe-mail: [email protected]: http://academics.smcvt.edu/jellis-monaghan

Page 2: St. Michaels College, Colchester, VT USA e-mail: jellis ...academics.smcvt.edu/jellis-monaghan/Talks/multivariable extension… · 1-partition 2-Partition 3-Partition. 4/10/08 Slide

4/10/08 Slide 2

Bridges and Loops

A bridge is an edge whose deletion separates the graph

A loop is an edge with both ends incident to the same vertex

bridgesNot a bridge

A loop

Page 3: St. Michaels College, Colchester, VT USA e-mail: jellis ...academics.smcvt.edu/jellis-monaghan/Talks/multivariable extension… · 1-partition 2-Partition 3-Partition. 4/10/08 Slide

4/10/08 Slide 3

e

Delete e

Contract eG

G-e

G/e

Deletion and contraction

Circuit vs cycle…

Page 4: St. Michaels College, Colchester, VT USA e-mail: jellis ...academics.smcvt.edu/jellis-monaghan/Talks/multivariable extension… · 1-partition 2-Partition 3-Partition. 4/10/08 Slide

4/10/08 Slide 4

Classical Dichromatic Polynomiala.k.a. Potts Model Partition Function

The classical dichromatic polynomial:

Potts Model Partition Function with q spins, and Hamiltonian

All edges have the same interaction energy in this model.If q and v are indeterminates rather than physical values, these

polynomials are the same (Fortuin & Kastelyn ’72).

( ) ( )

( )

( ); , A n k Ak A

A E G

Z G u v u v − +

= ∑

exp( ) 1v J Tκ= −

( )( ) ( , )i j

ij E Gh Jω δ σ σ

= − ∑

( ) ( )

( )

( )

( ); , expA n k Ak A

A E G states

hZ G q v q vTω

ωκ

− +

−⎛ ⎞= = ⎜ ⎟⎝ ⎠

∑ ∑

Page 5: St. Michaels College, Colchester, VT USA e-mail: jellis ...academics.smcvt.edu/jellis-monaghan/Talks/multivariable extension… · 1-partition 2-Partition 3-Partition. 4/10/08 Slide

4/10/08 Slide 5

Essential Properties• if e is not a loop.

• if GH is the disjoint union of G and H.

• if G consists of a single vertex and nloops.

(Whitney) Tutte ’67

( )( ; , ) ( ; , ) / ; ,Z G u v Z G e u v Z G e u v= − +

( ) ( ) ( ); , ; , ; ,Z GH u v Z G u v Z H u v=

( ; , ) ( 1)nZ G u v u v= +

These capture essential (local) physical properties, and are (defining) theoretical tools.

Page 6: St. Michaels College, Colchester, VT USA e-mail: jellis ...academics.smcvt.edu/jellis-monaghan/Talks/multivariable extension… · 1-partition 2-Partition 3-Partition. 4/10/08 Slide

4/10/08 Slide 6

Some Multivariable Dichromatic PolynomialsTraldi ‘89:

The interaction energy may depend on the edge

Fortuin & Kastelyn ‘72:

Doubly weighted, but we+ ve = 1, so equivalent to above.

( )( )

( ) ( )

( )

( )( ); , , ; ,A n k A k Ak Ae e

A E G A E Ge A e A

Z G t q q t v Z G q q v− +

⊆ ⊆∈ ∈

= → =∑ ∑∏ ∏v v

( ) ( )

( ); , ,

c

k Ae e

A E G e A e A

Z G q q v w⊆ ∈ ∈

= ∑ ∏ ∏v w

( )( ; , ) ( ; , ) / ; ,eZ G u Z G e u v Z G e u= − +v v v

( )( ; , , ) ( ; , , ) / ; , ,e eZ G u w Z G e u v Z G e u= − +v w v w v w

Page 7: St. Michaels College, Colchester, VT USA e-mail: jellis ...academics.smcvt.edu/jellis-monaghan/Talks/multivariable extension… · 1-partition 2-Partition 3-Partition. 4/10/08 Slide

4/10/08 Slide 7

A slight shift…Classical Tutte polynomial:

( ) ( ); , ( ; , ) \ ; ,T G x y T G e x y T G e x y= − +

( ); , i jT G x y x y=

Let e be an edge of G that is neither a bridge nor a loop. Then,

And if G consists of i bridges and j loops, then

( ) ( ) ( ); , 1 ; ,V k Gk G u vu v T G v Z G u vv

− +⎛ ⎞+ =⎜ ⎟⎝ ⎠

The Tutte polynomial is a translation of the dichromatic/Potts

Page 8: St. Michaels College, Colchester, VT USA e-mail: jellis ...academics.smcvt.edu/jellis-monaghan/Talks/multivariable extension… · 1-partition 2-Partition 3-Partition. 4/10/08 Slide

4/10/08 Slide 8

Fully ‘multivariablized’ Tutte polynomialZaslavsky ’92, Bollobás & Riordan ’99, (E-M &Traldi ’06)

( ),n nW E c α= nE

( )( )( )( )

/ , if is a bridge

, , if is a loop

/ , ( , ) else

e

e

e e

X W G e c e

W G c Y W G e c e

x W G e c y W G e c

⎧⎪

= −⎨⎪ + −⎩

Let , where is the edgeless graph on n vertices.

Each edge has four parameters associated with it: two ‘interaction energies’, a loop value, and a bridge value.

Page 9: St. Michaels College, Colchester, VT USA e-mail: jellis ...academics.smcvt.edu/jellis-monaghan/Talks/multivariable extension… · 1-partition 2-Partition 3-Partition. 4/10/08 Slide

4/10/08 Slide 9

Need to be careful about order…

( )( )

0

0

0

X y y X x Y Y x

Y x Y Y x x y y x

X x Y Y x x y y x

λ μ λ μ λ μ λ μ

ν λ μ λ μ λ μ λ μ

ν λ μ λ μ λ μ λ μ

− − + =

− − + =

− − + =

Need to have:

Necessary and sufficient to assure the function is well-defined, i.e. independent of the order of deletion and contraction.

vs.

Page 10: St. Michaels College, Colchester, VT USA e-mail: jellis ...academics.smcvt.edu/jellis-monaghan/Talks/multivariable extension… · 1-partition 2-Partition 3-Partition. 4/10/08 Slide

4/10/08 Slide 10

Dichromatic analogDon’t always get something analogous to the dichromatic, but do in all ‘reasonable’ situations, e.g. parameters in a field and not all zero, domain = all graphs.

In this case, with

( ) ( ) ( ) ( )

( ); , , ,

c

S n k Sk S k Ge e

S E G e S e S

Z G q r q r x y− +−

⊆ ∈ ∈

= ∑ ∏ ∏x y

, e e e e e eX x qy Y x ry= + = +

This gives a model with two degrees of freedom in the weights.

( )

( ) c

k Se e

S E G e S e S

q x y⊆ ∈ ∈

→ ∑ ∏ ∏

Page 11: St. Michaels College, Colchester, VT USA e-mail: jellis ...academics.smcvt.edu/jellis-monaghan/Talks/multivariable extension… · 1-partition 2-Partition 3-Partition. 4/10/08 Slide

4/10/08 Slide 11

The Martin, or circuit partition polynomial

These polynomials are generating functions for Eulerian k-partitions in a graph.

Martin ’77, cf Las Vergnas

Let G be an Eulerian graph (all vertices of even degree).

Let G be an Eulerian digraph (directed edges and Kirchhoff laws).

( ) ( )1 1

; ( ) , ; ( )k kk k

k k

J G x f G x j G x f G x= =

= =∑ ∑

G = An Euleriandigraph G :

An Eulerian 2-partition of G :

1-partition 2-Partition 3-Partition

Page 12: St. Michaels College, Colchester, VT USA e-mail: jellis ...academics.smcvt.edu/jellis-monaghan/Talks/multivariable extension… · 1-partition 2-Partition 3-Partition. 4/10/08 Slide

4/10/08 Slide 12

State Model and Recursive Forms

= xv → 1 + 1 + 1

v → 1 + 0 + 1

The Circuit Partition Polynomial for Oriented Graphs: (The weight is 1 for coherentstates, 0 else)

The Circuit Partition Polynomial

= x

( )( )

( )( )

( ) ( ); , ;c S c S

S states G S states G

J G x x j G x x∈ ∈

= =∑ ∑A graph state with 2 components

Multiple formulations facilitate theoretical techniques, e.g. induction.

Page 13: St. Michaels College, Colchester, VT USA e-mail: jellis ...academics.smcvt.edu/jellis-monaghan/Talks/multivariable extension… · 1-partition 2-Partition 3-Partition. 4/10/08 Slide

4/10/08 Slide 13

v → a + b + c

= x

The general form:

A multivariable, or weighted, version“On Transition Polynomials of 4-Regular Graphs” --F. Jaeger, 1987

( ) 3 2; , 2Q G x x xA x= + +

v → 1 + 1 + 0

= x

, get

+

+ + +

Then for G =

So

For example, if the weight system is:

A is the weight system specifying the coefficient for each transition.

Page 14: St. Michaels College, Colchester, VT USA e-mail: jellis ...academics.smcvt.edu/jellis-monaghan/Talks/multivariable extension… · 1-partition 2-Partition 3-Partition. 4/10/08 Slide

4/10/08 Slide 14

, where G is an arbitrary Eulerian graph, and W is a weight system which assigns a value in R (a ring with unit) to every pair of adjacent half edges in G. Then,

( ); ,N G W x

if pair bounds the same black face

if pair bounds the same white face else

b

W w

c

⎧⎪⎪= ⎨⎪⎪⎩

+ defabc + … = lmnx

E.g.

(New edge pairs have weight 1)

G =

4 3 2 2 2 2 2 2 ...b x b w x b c x= + + +

+ w2

+ …

+ c2b2

=

b2 b2 + w2 + c2

The generalized transition polynomialE-M 98, E-M & Sarmiento 02

Page 15: St. Michaels College, Colchester, VT USA e-mail: jellis ...academics.smcvt.edu/jellis-monaghan/Talks/multivariable extension… · 1-partition 2-Partition 3-Partition. 4/10/08 Slide

4/10/08 Slide 15

Overview of relationships

Parameterized TuttePolynomials

The ClassicalTutte Polynomial

The GeneralizedTransition Polynomial

For planar graphs via the medial graph

The Circuit Partition Polynomial

Page 16: St. Michaels College, Colchester, VT USA e-mail: jellis ...academics.smcvt.edu/jellis-monaghan/Talks/multivariable extension… · 1-partition 2-Partition 3-Partition. 4/10/08 Slide

4/10/08 Slide 16

The Tutte-transition connection

A Planar graph G Gm with the vertex faces colored black

Orient Gm so that black faces are to the left of each edge.

e

delete

contract

Then, with this orientation of Gm, ( ) ( ); ( ; 1, 1)k Gmj G x x t G x x= + +

Martin 77

Page 17: St. Michaels College, Colchester, VT USA e-mail: jellis ...academics.smcvt.edu/jellis-monaghan/Talks/multivariable extension… · 1-partition 2-Partition 3-Partition. 4/10/08 Slide

4/10/08 Slide 17

Same construction for multivariable case

v → √a √a + √b √b + 0

ewith weights a and b

The edge weights of G induce a weight system on mG

G a planar graphs with oriented medial graph mG

Then ( ) ( ); , ( ; , , , )k GmN G W x x Z G x x= v w

The multivariable Tutte & transition polynomials are alsorelated for planar graphs via the medial construction.

Page 18: St. Michaels College, Colchester, VT USA e-mail: jellis ...academics.smcvt.edu/jellis-monaghan/Talks/multivariable extension… · 1-partition 2-Partition 3-Partition. 4/10/08 Slide

4/10/08 Slide 18

Beyond Tutte: The Penrose Polynomial

N assimilates polynomials the Tutte polynomial doesn’t.

( ); , ( ; )mN G A x P G x=

v → 0 + 1 - 1

Defined for planar graphs and computed via the medial graph. Motivated by diagrams from tensor analysis, and surprisingly, for 3-regular planar graphs,

# 3-edge colorings ↔ 4 color theorem( ) ( )21;3 ; 2

4

V

P G P G−⎛ ⎞= − =⎜ ⎟⎝ ⎠

“Applications of Negative Dimensional Tensors”—R. Penrose, 1969

Page 19: St. Michaels College, Colchester, VT USA e-mail: jellis ...academics.smcvt.edu/jellis-monaghan/Talks/multivariable extension… · 1-partition 2-Partition 3-Partition. 4/10/08 Slide

4/10/08 Slide 19

The Kauffman bracket of knot theory

( )2 2 2 2; , ( ) [ ]LN G W a a a a K L− −+ = +

v+

→ 1/( √a √a) + √a √a + 0

v-

→ √a √a +1/( √a √a) + 0

The Kauffman bracket of a link L:

[ ] 1K =○[ ] 1K L∪ =○

Let GL be the signed, 2 face colored universe of a link L:

a + a-1

Page 20: St. Michaels College, Colchester, VT USA e-mail: jellis ...academics.smcvt.edu/jellis-monaghan/Talks/multivariable extension… · 1-partition 2-Partition 3-Partition. 4/10/08 Slide

4/10/08 Slide 20

Hopf Algebras—definition by exampleThe Binomial Bialgebra B

B is an infinite dimensional vector space over C with basis

B is an algebra, with (associative) multiplication given by:by

B is a coalgebra, with (coassociative) comultiplication given by:

B is a bialgebra, since the multiplication and the comultiplicationare compatible, i.e. the comultiplication is an algebra map:

, where

B is a Hopf algebra, with antipode:

{ }21, ,x x …

:m B B B⊗ → ( )r s r sm x x x +⊗ =

0

nn n r r

r

nx x x

r−

=

⎛ ⎞Δ = ⊗⎜ ⎟

⎝ ⎠∑

( )r s r sx x x xΔ ⋅ = Δ ⋅Δ ( ) ( )a b c d ac bd⊗ ⋅ ⊗ = ⊗

( )S x x= −

Page 21: St. Michaels College, Colchester, VT USA e-mail: jellis ...academics.smcvt.edu/jellis-monaghan/Talks/multivariable extension… · 1-partition 2-Partition 3-Partition. 4/10/08 Slide

4/10/08 Slide 21

is a locally finite partially ordered set with order relationΓ

( )( ) ( )( ), ,A W A G W G≤

( )W A ( )W G

Γ is a hereditary family

if A is an Eulerian subgraph of G, and if A has a weight system

that is inherited from

.

is also closed under disjoint union (direct product)

( ){ , }G WΓ =

Underlying Algebraic Structure

Γ

Page 22: St. Michaels College, Colchester, VT USA e-mail: jellis ...academics.smcvt.edu/jellis-monaghan/Talks/multivariable extension… · 1-partition 2-Partition 3-Partition. 4/10/08 Slide

4/10/08 Slide 22

:m Γ⊗Γ→ Γ

( )( ) ( )( )( ) ( )( ), , ,m G W G H W H GH W GH⊗ =

: Rμ →Γ ( ) ( ),r r Wμ = E

:Δ Γ→ Γ⊗Γ

( ) ( )( ) ( )( )1 1 2 2, , ,G W A W A A W AΔ = ⊗∑

: Rε Γ → ( )1 if

,0 else

GG Wε

=⎧= ⎨⎩

E

( ) ( ) ( )( )1 1, 1 ,PP PG W A A W A Aζ = −∑ … …

P

is an incidence Hopf algebra (Schmitt, 94).

by

Unit: by

Comultiplication: by

Counit: by

Antipode:

sum over all ordered partitions P of G into edge-disjoint Eulerian subgraphs.

Multiplication:

( ){ , }Rspan G WΓ =

Incidence Hopf Algebra

Page 23: St. Michaels College, Colchester, VT USA e-mail: jellis ...academics.smcvt.edu/jellis-monaghan/Talks/multivariable extension… · 1-partition 2-Partition 3-Partition. 4/10/08 Slide

4/10/08 Slide 23

Theorem: If we give the structure of a binomial

bialgebra, then is a Hopf algebra map.

The proof is straightforward combinatorics, but requires hereditary properties, and hence the pair weights and not just state weights.

[ ]R x

[ ]:N R xΓ→

N is a Hopf algebra map

Page 24: St. Michaels College, Colchester, VT USA e-mail: jellis ...academics.smcvt.edu/jellis-monaghan/Talks/multivariable extension… · 1-partition 2-Partition 3-Partition. 4/10/08 Slide

4/10/08 Slide 24

Structural identities

( ) ( )( ) ( )( )1 1 2 2; , ; , ; ,N G W x y N A W A x N A W A y+ =∑

( ) ( )( ); , , ;N G W x N G W xζ− =

II. (From the antipode)

I. (From the comultiplication)

These are powerful theoretic tools, particularly for inductive arguments.

Page 25: St. Michaels College, Colchester, VT USA e-mail: jellis ...academics.smcvt.edu/jellis-monaghan/Talks/multivariable extension… · 1-partition 2-Partition 3-Partition. 4/10/08 Slide

4/10/08 Slide 25

Identity I:Combinatorial interpretations for the Martin polynomials for allintegers (previously only known for –2, -1, 0, 1 in the oriented case, and –2, 0, 2 in the unoriented case) –E-M, also Bollobas.

Combinatorial interpretations for the diagonal Tutte polynomial (also derivatives) of a planar graph for all integers (previously –1, 3 were the only known non-trivial values) – E-M.

Identity II:Used to determine combinatorial interpretations for the Penrose polynomial for all negative integers (previously only known for positive integers)—Sarmiento, E-M&Sarmiento.

Some applications

Page 26: St. Michaels College, Colchester, VT USA e-mail: jellis ...academics.smcvt.edu/jellis-monaghan/Talks/multivariable extension… · 1-partition 2-Partition 3-Partition. 4/10/08 Slide

4/10/08 Slide 26

Jo Ellis-MonaghanSt. Michaels College, Colchester, VT USAe-mail: [email protected]: http://academics.smcvt.edu/jellis-monaghan