Possible quantum algorithms for the Bollobas-Riordan-Tutte polynomial of a ribbon graph

Possible Quantum Algorithms for the Bollobás-Riordan-Tutte Polynomial of a Ribbon Graph

Mario Vélez and Juan Ospina Logic and Computation Group Physical Engineering Program

School of Sciences and Humanities EAFIT University

{mvelez, jospina}@eafit.edu.co Abstract. Three possible quantum algorithms, for the computation of the Bollobás-Riordan-Tutte polynomial of a given ribbon graph, are presented and discussed. The first possible algorithm is based on the spanning quasi-trees expansion for generalized Tutte polynomials of generalized graphs and on a quantum version of the Binary Decision Diagram (BDD) for quasi-trees . The second possible algorithm is based on the relation between the Kauffman bracket and the Tutte polynomial; and with an application of the recently introduced Aharonov-Arad-Eban-Landau quantum algorithm. The third possible algorithm is based on the relation between the HOMFLY polynomial and the Tutte polynomial; and with an application of the Wocjan-Yard quantum algorithm. It is claimed that these possible algorithms may be more efficient that the best known classical algorithms. These three algorithms may have interesting applications in computer science at general or in computational biology and bio-informatics in particular. A line for future research based on the categorification project is mentioned. Keywords: Aharonov-Arad-Eban-Landau quantum algorithm, Potts problem, Tutte polynomial, Bollobás-Riordan-Tutte

polynomial, planar graph, ribbon graph, signed graph, Binary Decision Diagram..

1. Introduction The graph theory is a splendid source of very interesting computational hard problems. It is clear today that the algorithmic graph theory is urged by a more efficient computers than the classical computers. In fact with the advent of the quantum computing, such problems on graphs are being seriously considered as a possible candidates to be solved using quantum computers. Moreover very recently some quantum algorithms have been proposed inside the field of the quantum computational graph theory. More in detail, in the paper [1] the quantum versions for the Dijkstra and Prim algorithms are given and such quantum versions are more efficient than the well known original classical algorithms. From other side, in the paper [2], Aharonov-Arad-Eban-Landau are presenting certain quantum algorithm for the computation of the Tutte polynomial of a determined planar graph. The papers [1] and [2] are our source of inspiration and then in the present work we look by quantum algorithms for generalized Tutte polynomials. A very recently introduced generalized Tutte polynomial is the Bollobás-Riordan-Tutte(BRT) polynomial [3,4]. The BRT polynomial is the generalization of the Tutte polynomial [5] from the planar graphs to the ribbon graphs.

Quantum Information and Computation VI, edited by Eric J. Donkor, Andrew R. Pirich, Howard E. Brandt, Proc. of SPIE Vol. 6976, 69760O, (2008) · 0277-786X/08/$18 · doi: 10.1117/12.777401

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The more efficient algorithm actually known for the computation of the Tutte polynomial is implemented in the Maple 11 package GraphTheory using LISP, but unfortunately such algorithm is not efficient for the case of complete graphs with more than ten vertices. The same algorithm can be implemented via Haskell with some detectable better performance but again it fails in the case of graphs with more than ten vertices. Recently, a new algorithm was proposed which is efficient for graphs of moderate size (until 15 vertices) and which is based on the powerful computational notion of a BDD [6]. It is clear that a more effective algorithm is required for graphs with more that 15 vertices. As it was said a possible source of such required algorithms is placed in the field of the quantum computation and more specifically in the domain of the topological quantum computation.

2. Tutte and Bollobás-Riordan-Tutte polynomials In this section we collect the background about the Tutte polynomial and its generalizations such as the Bollobás-Riordan-Tutte (BRT) polynomial. The Tutte polynomial was originally introduced by W. T. Tutte [5] inside the context of the graph theory and subsequently was extended by Crapo [7] to the matroid theory. The exact evaluation of the Tutte polynomial and its generalizations is a #P-hard problem [8]. There are many definitions of the Tutte polynominal but the more standard is the original spanning tree expansion [5]:

( ) ( )( ; , ) i j

G

T G x y x yτ τ

τ ⊆

= ∑ , (1)

where G is a given graph, τ is a spanning tree of G, i(τ) is the number of internally active edges of τ , j(τ) is the number of externally active edges of τ, x is the variable of internal activity, y is the variable of external activity, and the sum is over all spanning trees of G. Other computational useful definition of the Tutte polynomial is [6]:

( )( ) ( )( ; , ) ( 1) ( 1) A AE A

A E

T G x y x y ρρ ρ −−

⊆

= − −∑ , (2)

where E is the edges set of G, ρ is the rank function of G, | A | is the cardinality of the subset A, and the sum is over all A of G. According with (1) and (2) the Tutte polynomial is a polynomial with two variables x and y, but recently certain multivariable generalizations of the Tutte polynomial have been discovered and the more celebrated is the corresponding to the Bollobás-Riordan-Tutte (BRT) polynomial [3,4] . The BRT polynomial is a polynomial on three or four variables which is a generalization of the Tutte polynomial from the planar graphs to the ribbon graphs, signed graphs and matroids. In the case of a ribbon graph, the BRT polynomial is defined as [3,4,9]:

( ) ( ) ( ) ( ) ( ) ( )

( )( ; , , ) r G r F n F k F bc F n F

F GR G x y z x y z− − +

∈Γ

= ∑ , (3)

where G is a ribbon graph with v(G) vertices, e(G) edges, k(G) connected components, with rank r(G)=v(G)-k(G) , nullity n(G)=e(G)-r(G) and bc(G) connected components for the boundary of the surface G. In (3), F is a spanning subgraph of G, Γ(G) is the set of spanning subgraphs of G, with |Γ(G)| = 2e(G); and the sum is over all the spanning subgraphs of G. In the case of a signed ribbon graph the BRT polynomial is defined as [9]:

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NflZ

ˆ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

ˆ( )

ˆ( ; , , ) r G r F s F n F s F k F bc F n F

F G

R G x y z x y z− + − − +

∈Γ

= ∑ , (4)

where G is signed ribbon graph with a given sign function and being

ˆ( ) ( )( )2

e F e G Fs F − −− −= , (5)

with ( )e F− denoting the number of negative edges of the spanning subgraph F. According with (4) and (5) the BRT for signed ribbon graphs is a Laurent polynomial in x1/2 , y1/2 and z. Some illustrative computations of the Tutte and BRT polynomials are as follows.

CG(10)

A Spanning Tree of CG(10)

Figure 1. The complete graph CG(10).

For example in the case of the complete graph with 10 vertices and 45 edges, GC(10), which is showed in the figure 1 jointly with a representative spanning tree, the explicit form of the corresponding Tutte polynomial is given by:

( )T , ,( )CG 10 x y 1477770 x2 y6 67284 x4 24220 y27 40320 x 40320 y + + + + =

720 x3 y14 386940 y x3 4379144 y10 3921351 x y6 15120 x3 y11 + + + + +

2787120 y14 932940 x y14 1726410 x2 y5 118124 x3 144060 y x4 109584 x2 + + + + + +

1672020 x2 y3 453510 x y16 154020 x3 y7 315 x2 y20 342090 x3 y5 + + + + + 743120 y3 245664 y2 425964 y21 546 x7 3068150 x y9 606816 y20 + + + + + +

30240 x4 y5 314928 x y 76860 x2 y14 3677772 x y5 2045820 x y3 + + + + +

533400 y x2 3718764 y12 4106172 y11 4536 x6 360 y26 x 485310 x2 y10 + + + + + +

3026625 x y4 1190490 x2 y7 662940 x y15 2106048 x y11 4482190 y9 + + + + + 4359300 y8 3969168 y7 1022316 x y2 579250 x3 y3 680610 x2 y9 + + + + +

1174740 x2 y2 1836660 x2 y4 578970 x3 y2 2585296 x y10 3507180 x y8 + + + + +

1527120 y4 3310272 y6 15330 y2 x5 917070 x2 y8 22449 x5 6930 y3 x5 + + + + + +

137550 x4 y2 1870560 y16 469875 x3 y4 1050 x4 y9 28812 y x5 93450 x4 y3 + + + + + +

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1266195 x y13 2447760 y5 838880 y19 1126720 y18 1471680 y17 + + + + +

215145 x2 y12 1287 y31 3265710 y13 2940 y x6 3826572 x y7 330960 x2 y11 + + + + + +

36 x8 1660281 x y12 235515 x3 y6 34275 x y21 41580 x2 y15 2314320 y15 + + + + + +

289740 y22 56700 x3 y9 297990 y17 x 15960 x4 y6 7350 x4 y7 190620 y23 + + + + + + 165 y33 112420 x y19 30660 x3 y10 121020 y24 54600 x4 y4 96300 x3 y8 + + + + + +

73932 y25 9 y35 12860 y28 20790 x2 y16 3300 y24 x 63945 x y20 + + + + + +

187460 y18 x 1260 x2 y19 132615 x2 y13 43308 y26 210 x4 y10 10 y28 x + + + + + +

3003 y30 252 x5 y6 7920 y23 x 2520 x5 y4 3780 y18 x2 9450 y17 x2 y36 + + + + + + + 495 y32 1008 x5 y5 17160 y22 x 3150 x4 y8 6720 x3 y12 630 y2 x6 + + + + + +

1200 y25 x 45 y34 2520 x3 y13 45 x2 y21 210 y3 x6 6435 y29 80 y27 x + + + + + + +

120 x3 y15 120 y x7 x9 + + +

By direct computation is obtained the total number of spanning trees of CG(10)

= ( )T , ,( )CG 10 = x 1 = y 1 100000000 , and the total number of forests of CG(10) is:

= ( )T , ,( )CG 10 = x 2 = y 1 205608536 Now, in the case of the ribbon graph depicted in the figure 2, jointly with a possible riemman surface on which it is inscribed [10],

Figure 2. A ribbon graph and one possible Riemann surface over which is depicted [10]

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4\

41+y+A+A_B+B_ + 2yA+A2 B+ + 24A+B+B+ !4AA_B_ + nA_BB_ + +

+ + v+AiB + z+ABt + x+4B+B.

the corresponding BRT polynomial is computed as:

.

Another example of BRT polynomial for the ribbon graph showed in the figure 3 [9],

Figure 3. A ribbon graph [9]

is [9]:

A final example of a BRT polynomial in four variables corresponding to the signed graph of figure 4 [11] :

Figure 4. Signed graph [11]

has the form [11]

As it was said before, the problem of the exact computation of the Tutte and BRT polynomial is a hard problem which can not be solved efficiently using classical computers but it is expected that such problem may be solved efficiently using quantum computers but at least in an approximate form.

3. The Aharonov-Arad-Eban-Landau Algorithm Some problems about graphs have been treated in the literature using quantum algorithms [1,2], but only recently the problem of computing the Tutte polynomial and its generalizations have been considered from the perspective of quantum computation. As it was said from the beginning, in the work [2] a quantum algorithm for the Tutte polynomial was proposed inside the project to find a solution for the legendary Potts problem in statistical physics. In this section we give a sketch of this algorithm which is abbreviated as AAEL algorithm.

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The basic equations of the AAEL algorithm are as follows. Full details can be obtained in the original work [2] and also in [12]. The partition function for the Potts model is defined as:

( )HKT

PottsZ eα

α

−= ∑ , (6)

where H(α) is the energy of the configuration with label α, K is the Boltzmann constant and T is the temperature. From other side, the multivariable Tutte polynomial or dichromatic polynomial is defined as [2]

( )( ; , ) k Ae

A E e A

Z G q v q v⊆ ∈

= ∑ ∏r, (7)

where G(V,E) is a given graph, vr is vector of weights and q is a “quantum” parameter. The more standard Tutte polynomial with two variables given by (2) is represented now as:

( )( ; , ) ( 1) ( 1) ( ;( 1)( 1), 1)Vk ET G x y x y Z G x y y−−= − − − − −uuuur

, (8)

and the important relation among the multivariable Tutte polynomial (7) and the Potts partition function (6) is given by

( ; , )PottsZ Z G q v=r

, (9)

where q and vr are now physical parameters of the Potts model. Specifically in the AAEL quantum algorithm the aim is to compute the multivariable Tutte polynomial or dichromatic polynomial (7), for a given graph; and then to compute the partition function of the Potts model corresponding to such graph. The strategy in the AAEL algorithm is exploit a relation between the multivariable Tutte polynomial (7) and the Kauffman bracket for a certain graph resulting from a determined link associated with the original graph. Such relation reads:

2( ; , ) ( ; , )VGK L d u d Z G d du−=

r r, (10)

where the Kauffman bracket is defined as

( )( ; , ) s s eG e

sK L d u d u= ∑ ∏

r. (11)

All these equations are purely mathematical but the quantum effects appear when the following equation is introduced:

( ; , ) 1 1GK L d u Q=r

, (12)

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which is a relation between the Kauffman bracket and the quantum expected value of the quantum observable Q being this latter defined as

( ( ))GQ Lρ= Ψ , (13) where ρ is a representation not necessarily unitary, for the virtual Temperley-Lieb algebra [2]. With the equations (6)-(13) the AAEL algorithm can be presented in the following way [2].

1. Introduce a graph G as the input. 2. Obtain the link LG associated with the graph G. 3. Write the link LG as a word in the Artin braid algebra. 4. Make a representation of the Artin braid algebra inside the virtual Temperley-Lieb algebra. 5. Obtain the virtual Temperley-Lieb representation of the braid-word for LG. 6. Determine the quantum charge Q according with (13)

7. Use a standard quantum computer to determine 1 1Q .

8. Apply (12) to obtain the Kauffman bracket for LG. 9. Apply (11) to obtain 2( ; , )Z G d du

r.

10. Apply (9) to obtain ZPottss. 11. Apply (8) to obtain the Tutte polynomial T(G; x , y ).

The AAEL algorithm gives an approximation of the Tutte polynomial for almost any pair of values for (x , y) but the algorithm does not provide the explicit or symbolic form of the Tutte polynomial. In all case but at least in theory, the AAEL algorithm is more efficient than the more efficient classical algorithms when very large graphs are involved. More commentaries about the AAEL algorithm can be observed in [12].

4. Quantum Algorithms for the BRT Polynomial In this section three possible quantum algorithms for the computation of the BRT polynomial of a given ribbon graph are presented. The first algorithm is based on the quasi-tree expansion for the BRT polynomial [10] and on the quantum computing of the Binary Decision Diagram [6] corresponding to the set of all quasi-trees for the considered ribbon graph. The second algorithm is based on the relationship among the BRT polynomial and the Kauffman bracket for the particular link associated with the given ribbon graph [9] and doing use of a modified version of the AAEL algorithm. The third algorithm is based on the relation between the BRT polynomial and the HOMFLY polynomial [13] and using a modified version of the WJ algorithm[14].

4.1. First Algorithm: using the Quantum BDD for quasi-trees In the report [6], a very speed classical algorithm is given for the exact computation of the Tutte polynomial for graphs of moderate size. Such algorithm is based on the notion of a Binary Decision Diagram (BDD) for Boolean functions. The BDD contains in classified form the full set of spanning trees of a graph G, over which the sum in the equation (1) is realized. Now for the BRT polynomial the quasi-tree expansion is given by [10]:

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( )( ( )) ( ( )) ( ( )) ( ( ))

( )( ; , , ) (1 ) ( ; ,1 )Fn D F k D F bc D F n D F

FF G

R G x y z y z y T G x yzΕ− +

∈Γ

= + +∑ , (14)

where F is now a quasi-tree given by a ordered chord diagram, being D(F) the subgraph of internally inactive edges in F, being E(F) the set of externally active edges in G – F, and being GF the graph whose vertices are the components of D(F) and whose edges are the internally active edges of F. With these elements a possible quantum algorithm for the BRT polynomial using the quantum version of the BDD for quasi-trees can be depicted as the figure 5 shows.

Figure 5. The first quantum algorithm for the BRT polynomial

In words the figure 5, reads:

1. Introduce a ribbon graph as the input. 2. Run the quantum version of the BDD for quasi-trees. 3. Classify the full set of Spanning quasi- trees of the inputted ribbon graph. 4. Assign the corresponding polynomials to the equivalence classes previously obtained according with (14). 5. Make the sum to obtain the BRT polynomial according with (14). 6. Compute the number of quasi-forests for the inputted ribbon graph.

As it was said the classical algorithm with classical BDD [6] is more efficient that all other known classical algorithms, and for this reason the quantum algorithm in figure 5 is more efficient than any classical algorithm based on the quasi-tree expansion for the BRT polynomial given by (14).

4.2. Second Algorithm: using the AAEL Quantum Algorithm

Quantum BDD for quasi-trees

Full Set of quasi-Trees

Classifier

Equivalence Classes

Bollobás-Riordan-Tutte Polynomial

Number of Quasi-Forests

Polynomials Assigner

Summation

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A second possible algorithm for the BRT polynomial of a given ribbon graph can be obtained via an adaptation of the previously presented AAEL algorithm [2]. A key ingredient in the AAEL algorithm is the equation (10) which gives a relation between the Tutte polynomial and the Kauffman bracket. A similar relation exists between the BRT polynomial and the Kauffman bracket and this relation reads [9]:

( ) ( ) ( ) 1 1( ; , , ) ( ; , , )r G n G k GL

Bd AdK L A B d A B d R GA B d

−= , (15)

where GL is the ribbon graph associated with an alternating virtual link diagram L [9]. With these elements an adaptation of the AAEL algorithm for the case of the BRT polynomial, can be obtained and such possible quantum algorithm is depicted in the figure 6.

Figure 6. The second quantum algorithm for the BRT polynomial. The figure 6 can be translated in words as follows:

1. Introduce a ribbon graph as the input. 2. Convert the ribbon graph in a virtual diagram of a specific link. 3. Translate the obtained link in a virtual-braid-word. 4. Make a representation of the virtual braid algebra in the virtual Temperley-Lieb algebra. 5. Translate the virtual-braid-word previously obtained, in terms of the generators of the virtual Temperley-Lieb

algebra. 6. Compute the quantum mean value of the virtual Temperley-Lieb operator obtained previously. 7. Compute the Kauffman bracket via a little modification of the equation (12). 8. Compute the BRT polynomial using (15). 9. Compute the number of forests of the inputted ribbon graph.

4.3. Third Algorithm: using the Wocjan-Yard Algorithm A third quantum algorithm for the BRT polynomial of a ribbon graph can be obtained using the relation between the BRT polynomial and the HOMFLY polynomial of the associated link [13]; and using a modified version of the WY quantum algorithm [14] for the computation of the HOMFLY polynomial. The needed relation between the BRT polynomial and the HOMFLY polynomial is [13]:

Topological Converter

Knot or Link

Modified AAEL Algorithm

Kauffman Bracket

Algebraic Converter

BRT Polynomial

Number Of Forests

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1( ) 1 ( ) 2 ( ) 1 2

2 1

1( ( ); , ) ( ) ( ) ( 1) ( ; 1, , )v G e G k Gy x x yH L G x y x R G xxy x xy x x

− −−

−= − −

−, (16)

where G is a ribbon graph and H(L(G); x ,y) is the HOMFLY polynomial for the link L(G) associated with G [13]. In the quantum WJ algorithm [14], the HOMFLY polynomial is computed via the Markov trace for the representations of the braid algebra in the Iwahori-Hecke algebra [14], being this last algebra certain generalization of the Temperley-Lieb algebra used in the AAEL algorithm [2]. With these elements a third possible quantum algorithm for BRT polynomial is depicted in the figure 7.

Figure 7. The third quantum algorithm for the BRT polynomial.

The figure 7 can be translated in words as follows:

1. Introduce a ribbon graph as the input. 2. Convert the ribbon graph in a virtual diagram of a specific link. 3. Translate the obtained link in a virtual-braid-word. 4. Make a representation of the virtual braid algebra in the virtual Iwahori-Hecke algebra. 5. Translate the virtual-braid-word previously obtained, in terms of the generators of the virtual Iwahori-Hecke

algebra. 6. Compute the quantum trace of the virtual Iwahori-Hecke operator obtained previously. 7. Obtain the HOMFLY polynomial in terms of the quantum trace previously computed. 8. Compute the BRT polynomial using (16). 9. Compute the number of forests of the inputed ribbon graph.

It is necessary to remark here that the first possible algorithm of the figure 5, gives the exact and explicit or analytical or symbolic form of the BRT polynomial in a similar way than the Maple package “ GraphTheory” gives the Tutte polynomial of the complete graph with 10 vertices and 45 edges , CG(10). The other two possible algorithms showed in figures 6 and 7, give approximate numerical evaluations for the BRT polynomial at any pair of values (x , y). More concretely the first algorithm (figure 5) gives exact symbolic results for ribbon graphs of moderate size, and the other two algorithms (figure 6, figure 7) give approximate numerical results for ribbon graphs of large size. It is claimed here that these three possible algorithms given in the figures 5, 6 and 7, are more efficient than any classical algorithm oriented to the computation of the BRT polynomial of a ribbon graph. Finally it is worthwhile to comment that similar algorithms are possible for the computation of the BRT polynomial with four variables corresponding to signed ribbon graphs and matroids.

Topological Converter

Knot or Link

WY Algorithm

HOMFLY Polynomial

Algebraic Converter

BRT Polynomial

Number Of Forests

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5. Conclusions Initially the topological quantum computation was directed by the new advances in algebraic topology and quantum algebras but subsequently the mathematical background was shifted to the geometric topology and combinatorics particularly in the concerning with braid theory, knot theory and graph theory . In this work some applications of the topological quantum computation in graph theory were presented. Specifically some quantum algorithms for the computation of the Bollobas-Riordan-Tutte polynomial of a ribbon graph were proposed. Such algorithms were obtained as generalizations of the recently proposed Aharonov-Arad-Ebal-Landau algorithm for the computation of Tutte polynomial of a planar graph. Both the AAEL algorithm as the algorithms proposed here, are claimed as more efficient than the more powerful classical algorithms and all these quantum algorithms are viewed as forward steps to the solution of the Potts problem. From other side the presented algorithms can be useful in quantum computational graph theory in general and in computational biology and bio-informatics in particular, given in the mentioned bio-disciplines appear interesting and hard problems about graphs. A possible line of future research is inspired in the project of categorification according to which all invariant polynomials such as Jones, Tutte, HOMFLY and BRT can be derived as the Euler characteristic of a certain homology complex associated with the link or graph under consideration. We hope to have the opportunity to exploit this new computational strategy in our next work.. This work was supported by EAFIT University .

6. References [1] M. Heiligman, Quantum Algorithms for Lowest Weight Paths and Spanning Trees in Complete Graphs, arXiv:quant-ph/0303131v1, 2003. [2] D. Aharonov, I. Arad, E. Eban, Z. Landau, Polynomial Quantum Algorithms for Additive approximations of the Potts model and other Points of the Tutte Plane, arXiv:quant-ph/0702008v1, Presented at QIP 2007, Brisbane, Australia. [3] B. Bollobás, O. Riordan. A polynomial invariant of graphs on orientable surfaces. Proc. Londo Math. Soc. (3), 83 (3): 523-531, 2001. [4] B. Bollobás, O. Riordan. A polynomial of graphs on surfaces. Math. Ann., 323(1): 81-96, 2002. [5] W. Tutte. A contribution to the theory of chromatic polynomials. Canadian J. Math. 6 , 80-91, 1954. [6] K. Sekine, H. Imai, S. Tani, Computing the Tutte Polynomial of a Graph and the Jones Polynomial of an Alternating Link of Moderate Size, Technical Report 95-06, 1995. www.is.s.u-tokyo.ac.jp/tech-reports/FILES.html [7] Henry H. Crapo. The Tutte polynomial. Aequationes Mathematicae, 3, 221-229, 1969. [8] F. Jaeger, D.L. Vertigan and D.J.A. Welsh. On the computational complexity of the Jones and Tutte polynomials. Mathematical Proceedings of the Cambridge Philoso-Phical Society, 108, 35-53, 1990. [9] S. Chmutov, I. Pak, The Kauffman bracket of virtual links and the Bollobás-Riordan polynomial, arXiv:math/0609012v1[math.GT], 2006. To be published in: Moscow Mathematical Journal. [10] A. Champanerkar, I. Kofman, N. Stoltzfus, Quasi-tree expansion for the Bollobás-Riordan-Tutte polynomial, arXiv: 0705.3458v1[math.CO], 2007.

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[11] Y. Diao, G. Hetyei, K. Hinson. Tutte Polynomials of Tensor Products of Signed Graphs and their Applications in Knot Theory, arXiv:math/0702328v1, 2007. [12] J. Geraci, D. A. Lidar, On the Exact evaluation of Certain Instances of the Potts Partition Function by Quantum Computers, arXiv: quant-ph/ 0703023v1, 2007. [13] I. Moffat, Knot invariants and the Bollobás-Riordan polynomial of embedded Graphs, arXiv:math/0605466v2[math.CO], 2006, to appear in European Journal of Combinatorics. [14] P. Wocjan, J. Yard, The Jones polynomial: quantum algorithms and applications in quantum complexity theory, arXiv:quant-ph/0603069v3, 2006.

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