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 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 CATEGORIFICATION OF THE DICHROMATIC POLYNOMIAL FOR GRAPHS

2008 ◽  
Vol 17 (01) ◽  
pp. 31-45 ◽  
Author(s):  
MARKO STOŠIĆ
Keyword(s):  
Positive Integer ◽  
Euler Characteristic ◽  
Jones Polynomial ◽  
Chain Complex ◽  
Tutte Polynomial ◽  
A Chain ◽  
Alternating Link ◽  
Dichromatic Polynomial

For each graph and each positive integer n, we define a chain complex whose graded Euler characteristic is equal to an appropriate n-specialization of the dichromatic polynomial. This also gives a categorification of n-specializations of the Tutte polynomial of graphs. Also, for each graph and integer n ≤ 2, we define the different one-variable n-specializations of the dichromatic polynomial, and for each polynomial, we define graded chain complex whose graded Euler characteristic is equal to that polynomial. Furthermore, we explicitly categorify the specialization of the Tutte polynomial for graphs which corresponds to the Jones polynomial of the appropriate alternating link.

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.

scholarly journals Zeros of Jones Polynomials of Graphs

10.37236/4627 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Fengming Dong ◽  
Xian'an Jin
Keyword(s):  
Potts Model ◽  
Upper Bound ◽  
Maximum Degree ◽  
Jones Polynomial ◽  
Tutte Polynomial ◽  
Plane Graph ◽  
Partition Functions ◽  
Free Interval ◽  
Jones Polynomials ◽  
Alternating Link

In this paper, we introduce the Jones polynomial of a graph $G=(V,E)$ with $k$  components as the following specialization of the Tutte polynomial:$$J_G(t)=(-1)^{|V|-k}t^{|E|-|V|+k}T_G(-t,-t^{-1}).$$We first study its basic properties and determine certain extreme coefficients. Then we prove that $(-\infty, 0]$ is a zero-free interval of Jones polynomials of connected bridgeless graphs while for any small $\epsilon>0$ or large $M>0$, there is a zero of the Jones polynomial of a plane graph in $(0,\epsilon)$, $(1-\epsilon,1)$, $(1,1+\epsilon)$ or $(M,+\infty)$. Let $r(G)$ be the maximum moduli of zeros of $J_G(t)$. By applying Sokal's result on zeros of Potts model partition functions and Lucas's theorem, we prove that\begin{eqnarray*}{q_s-|V|+1\over |E|}\leq r(G)<1+6.907652\Delta_G\end{eqnarray*}for any connected bridgeless and loopless graph $G=(V,E)$ of maximum degree $\Delta_G$ with $q_s$ parallel classes. As a consequence of the upper bound, X.-S. Lin's conjecture holds if the positive checkerboard graph of a connected alternating link has a fixed maximum degree and a sufficiently large number of edges.

The Computational Complexity of the Tutte Plane: the Bipartite Case

1992 ◽  
Vol 1 (2) ◽  
pp. 181-187 ◽  
Author(s):  
D. L. Vertigan ◽  
D. J. A. Welsh
Keyword(s):  
Computational Complexity ◽  
Ising Model ◽  
Partition Function ◽  
Polynomial Time ◽  
Spanning Trees ◽  
Tutte Polynomial ◽  
Wide Range ◽  
Number Of Spanning Trees ◽  
Bipartite Case ◽  
Complete Characterisation

Along different curves and at different points of the (x, y)-plane the Tutte polynomial evaluates a wide range of quantities. Some of these, such as the number of spanning trees of a graph and the partition function of the planar Ising model, can be computed in polynomial time, others are # P-hard. Here we give a complete characterisation of which points and curves are easy/hard in the bipartite case.

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.

Jones polynomial invariants for knots and satellites

1991 ◽  
Vol 109 (1) ◽  
pp. 83-103 ◽  
Author(s):  
H. R. Morton ◽  
P. Strickland
Keyword(s):  
Quantum Group ◽  
Linear Combination ◽  
Simple Formula ◽  
Jones Polynomial ◽  
Polynomial Invariants ◽  
Satellite Link ◽  
Representation Ring ◽  
Parallel Links ◽  
Jones Polynomials ◽  
Special Case

AbstractResults of Kirillov and Reshetikhin on constructing invariants of framed links from the quantum group SU(2)q are adapted to give a simple formula relating the invariants for a satellite link to those of the companion and pattern links used in its construction. The special case of parallel links is treated first. It is shown as a consequence that any SU(2)q-invariant of a link L is a linear combination of Jones polynomials of parallels of L, where the combination is determined explicitly from the representation ring of SU(2). As a simple illustration Yamada's relation between the Jones polynomial of the 2-parallel of L and an evaluation of Kauffman's polynomial for sublinks of L is deduced.

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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