scholarly journals Graphs, Links, and Duality on Surfaces

2010 ◽  
Vol 20 (2) ◽  
pp. 267-287 ◽  
Author(s):  
VYACHESLAV KRUSHKAL
Keyword(s):  
Planar Graphs ◽  
Jones Polynomial ◽  
Tutte Polynomial ◽  
Natural Duality ◽  
Polynomial Invariant ◽  
Topological Duality ◽  
Ribbon Graphs ◽  
Virtual Links ◽  
Graphs On Surfaces ◽  
Tutte Polynomials

We introduce a polynomial invariant of graphs on surfaces,PG, generalizing the classical Tutte polynomial. Topological duality on surfaces gives rise to a natural duality result forPG, analogous to the duality for the Tutte polynomial of planar graphs. This property is important from the perspective of statistical mechanics, where the Tutte polynomial is known as the partition function of the Potts model. For ribbon graphs,PGspecializes to the well-known Bollobás–Riordan polynomial, and in fact the two polynomials carry equivalent information in this context. Duality is also established for a multivariate version of the polynomialPG. We then consider a 2-variable version of the Jones polynomial for links in thickened surfaces, taking into account homological information on the surface. An analogue of Thistlethwaite's theorem is established for these generalized Jones and Tutte polynomials for virtual links.

scholarly journals A Polynomial Invariant and Duality for Triangulations

10.37236/4162 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Vyacheslav Krushkal ◽  
David Renardy
Keyword(s):  
Planar Graphs ◽  
Essential Feature ◽  
Dual Graph ◽  
Tutte Polynomial ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Ribbon Graphs ◽  
Higher Dimensional ◽  
Classical Invariant ◽  
Dimension 2

The Tutte polynomial ${T}_G(X,Y)$ of a graph $G$ is a classical invariant, important in combinatorics and statistical mechanics. An essential feature of the Tutte polynomial is the duality for planar graphs $G$, $T_G(X,Y) = {T}_{G^*}(Y,X)$ where $G^*$ denotes the dual graph. We examine this property from the perspective of manifold topology, formulating polynomial invariants for higher-dimensional simplicial complexes. Polynomial duality for triangulations of a sphere follows as a consequence of Alexander duality. The main goal of this paper is to introduce and begin the study of a more general $4$-variable polynomial for triangulations and handle decompositions of orientable manifolds. Polynomial duality in this case is a consequence of Poincaré duality on manifolds. In dimension 2 these invariants specialize to the well-known polynomial invariants of ribbon graphs defined by B. Bollobás and O. Riordan. Examples and specific evaluations of the polynomials are discussed.

Types of embedded graphs and their Tutte polynomials

2019 ◽  
Vol 169 (2) ◽  
pp. 255-297 ◽  
Author(s):  
STEPHEN HUGGETT ◽  
IAIN MOFFATT
Keyword(s):  
Systematic Approach ◽  
Tutte Polynomial ◽  
Embedded Graphs ◽  
Graph Polynomials ◽  
Ribbon Graphs ◽  
Duality Relations ◽  
Graphs On Surfaces ◽  
Tutte Polynomials

AbstractWe take an elementary and systematic approach to the problem of extending the Tutte polynomial to the setting of embedded graphs. Four notions of embedded graphs arise naturally when considering deletion and contraction operations on graphs on surfaces. We give a description of each class in terms of coloured ribbon graphs. We then identify a universal deletion-contraction invariant (i.e., a ‘Tutte polynomial’) for each class. We relate these to graph polynomials in the literature, including the Bollobás–Riordan, Krushkal and Las Vergnas polynomials, and give state-sum formulations, duality relations, deleton-contraction relations, and quasi-tree expansions for each of them.

scholarly journals THISTLETHWAITE'S THEOREM FOR VIRTUAL LINKS

2008 ◽  
Vol 17 (10) ◽  
pp. 1189-1198 ◽  
Author(s):  
SERGEI CHMUTOV ◽  
JEREMY VOLTZ
Keyword(s):  
Planar Graph ◽  
Jones Polynomial ◽  
Tutte Polynomial ◽  
Virtual Links ◽  
Higher Genus

The celebrated Thistlethwaite theorem relates the Jones polynomial of a link with the Tutte polynomial of the corresponding planar graph. We give a generalization of this theorem to virtual links. In this case, the graph will be embedded into a (higher genus) surface. For such graphs we use the generalization of the Tutte polynomial discovered by Bollobás and Riordan.

On the computational complexity of the Jones and Tutte polynomials

1990 ◽  
Vol 108 (1) ◽  
pp. 35-53 ◽  
Author(s):  
F. Jaeger ◽  
D. L. Vertigan ◽  
D. J. A. Welsh
Keyword(s):  
Computational Complexity ◽  
Jones Polynomial ◽  
Tutte Polynomial ◽  
Wide Range ◽  
Tutte Polynomials ◽  
Special Points ◽  
Alternating Link ◽  
Special Case

AbstractWe show that determining the Jones polynomial of an alternating link is #P-hard. This is a special case of a wide range of results on the general intractability of the evaluation of the Tutte polynomial T(M; x, y) of a matroid M except for a few listed special points and curves of the (x, y)-plane. In particular the problem of evaluating the Tutte polynomial of a graph at a point in the (x, y)-plane is #P-hard except when (x − 1)(y − 1) = 1 or when (x, y) equals (1, 1), (−1, −1), (0, −1), (−1, 0), (i, −i), (−i, i), (j, j2), (j2, j) where j = e2πi/3

scholarly journals TUTTE POLYNOMIALS OF TENSOR PRODUCTS OF SIGNED GRAPHS AND THEIR APPLICATIONS IN KNOT THEORY

2009 ◽  
Vol 18 (05) ◽  
pp. 561-589 ◽  
Author(s):  
Y. DIAO ◽  
G. HETYEI ◽  
K. HINSON
Keyword(s):  
Tensor Product ◽  
Knot Theory ◽  
Jones Polynomial ◽  
Tensor Products ◽  
Tutte Polynomial ◽  
Signed Graphs ◽  
Substitution Rule ◽  
Simple Substitution ◽  
Jones Polynomials ◽  
Tutte Polynomials

It is well-known that the Jones polynomial of an alternating knot is closely related to the Tutte polynomial of a special graph obtained from a regular projection of the knot. Relying on the results of Bollobás and Riordan, we introduce a generalization of Kauffman's Tutte polynomial of signed graphs for which describing the effect of taking a signed tensor product of signed graphs is very simple. We show that this Tutte polynomial of a signed tensor product of signed graphs may be expressed in terms of the Tutte polynomials of the original signed graphs by using a simple substitution rule. Our result enables us to compute the Jones polynomials of some large non-alternating knots. The combinatorics used to prove our main result is similar to Tutte's original way of counting "activities" and specializes to a new, perhaps simpler proof of the known formulas for the ordinary Tutte polynomial of the tensor product of unsigned graphs or matroids.

Relative Tutte Polynomials for Coloured Graphs and Virtual Knot Theory

2009 ◽  
Vol 19 (3) ◽  
pp. 343-369 ◽  
Author(s):  
Y. DIAO ◽  
G. HETYEI
Keyword(s):  
Knot Theory ◽  
Jones Polynomial ◽  
Tutte Polynomial ◽  
Ribbon Graph ◽  
Virtual Knot ◽  
Tree Expansion ◽  
The Face ◽  
Tutte Polynomials ◽  
Bracket Polynomial ◽  
Virtual Knot Theory

We introduce the concept of a relative Tutte polynomial of coloured graphs. We show that this relative Tutte polynomial can be computed in a way similar to the classical spanning tree expansion used by Tutte in his original paper on this subject. We then apply the relative Tutte polynomial to virtual knot theory. More specifically, we show that the Kauffman bracket polynomial (and hence the Jones polynomial) of a virtual knot can be computed from the relative Tutte polynomial of its face (Tait) graph with some suitable variable substitutions. Our method offers an alternative to the ribbon graph approach, using the face graph obtained from the virtual link diagram directly.

scholarly journals Relative Tutte Polynomials of Tensor Products of Coloured Graphs

2013 ◽  
Vol 22 (6) ◽  
pp. 801-828
Author(s):  
Y. DIAO ◽  
G. HETYEI
Keyword(s):  
Tensor Product ◽  
Jones Polynomial ◽  
Tensor Products ◽  
Tutte Polynomial ◽  
Product Formula ◽  
Virtual Knot ◽  
Tutte Polynomials ◽  
Product Of Graphs

The tensor product (G1,G2) of a graph G1 and a pointed graph G2 (containing one distinguished edge) is obtained by identifying each edge of G1 with the distinguished edge of a separate copy of G2, and then removing the identified edges. A formula to compute the Tutte polynomial of a tensor product of graphs was originally given by Brylawski. This formula was recently generalized to coloured graphs and the generalized Tutte polynomial introduced by Bollobás and Riordan. In this paper we generalize the coloured tensor product formula to relative Tutte polynomials of relative graphs, containing zero edges to which the usual deletion/contraction rules do not apply. As we have shown in a recent paper, relative Tutte polynomials may be used to compute the Jones polynomial of a virtual knot.

ON TANGLES AND MATROIDS

2005 ◽  
Vol 14 (07) ◽  
pp. 919-929 ◽  
Author(s):  
STEPHEN HUGGETT
Keyword(s):  
Tensor Product ◽  
Planar Graph ◽  
Planar Graphs ◽  
Knot Theory ◽  
Tutte Polynomial ◽  
The Other ◽  
Link Diagram ◽  
Polynomial Invariant ◽  
Link Diagrams ◽  
Alternating Link

Given matroids M and N there are two operations M ⊕2 N and M ⊗ N. When M and N are the cycle matroids of planar graphs these operations have interesting interpretations on the corresponding link diagrams. In fact, given a planar graph there are two well-established methods of generating an alternating link diagram, and in each case the Tutte polynomial of the graph is related to a polynomial invariant (Jones or Homfly) of the link. Switching from one of these methods to the other corresponds in knot theory to tangle insertion in the link diagrams, and in combinatorics to the tensor product of the cycle matroids of the graphs.

scholarly journals The Tutte polynomial of symmetric hyperplane arrangements

2020 ◽  
Vol 29 (03) ◽  
pp. 2050004
Author(s):  
Hery Randriamaro
Keyword(s):  
Hyperplane Arrangement ◽  
Dual Graph ◽  
Tutte Polynomial ◽  
Hyperplane Arrangements ◽  
Weyl Groups ◽  
Reflection Groups ◽  
Tutte Polynomials ◽  
One Year ◽  
The One ◽  
Definition Of

The Tutte polynomial is originally a bivariate polynomial which enumerates the colorings of a graph and of its dual graph. Ardila extended in 2007 the definition of the Tutte polynomial on the real hyperplane arrangements. He particularly computed the Tutte polynomials of the hyperplane arrangements associated to the classical Weyl groups. Those associated to the exceptional Weyl groups were computed by De Concini and Procesi one year later. This paper has two objectives: On the one side, we extend the Tutte polynomial computing to the complex hyperplane arrangements. On the other side, we introduce a wider class of hyperplane arrangements which is that of the symmetric hyperplane arrangements. Computing the Tutte polynomial of a symmetric hyperplane arrangement permits us to deduce the Tutte polynomials of some hyperplane arrangements, particularly of those associated to the imprimitive reflection groups.

scholarly journals A study about the Tutte polynomials of benzenoid chains

2017 ◽  
Vol 5 (1) ◽  
pp. 28-32
Author(s):  
Abdulgani Sahin
Keyword(s):  
Tutte Polynomial ◽  
Signed Graphs ◽  
Tutte Polynomials

Abstract The Tutte polynomials for signed graphs were introduced by Kauffman. In 2012, Fath-Tabar, Gholam-Rezaeı and Ashrafı presented a formula for computing Tutte polynomial of a benzenoid chain. From this point on, we have also calculated the Tutte polynomials of signed graphs of benzenoid chains in this study.

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