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.

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 The Jones polynomial: quantum algorithms and applications in quantum complexity theory

2008 ◽  
Vol 8 (1&2) ◽  
pp. 147-180
Author(s):  
P. Wocjan ◽  
J. Yard
Keyword(s):  
Quantum Computation ◽  
Braid Group ◽  
Quantum Algorithms ◽  
Primitive Root ◽  
Jones Polynomial ◽  
Tutte Polynomial ◽  
Roots Of Unity ◽  
Complete Problems ◽  
Primitive Roots ◽  
Quantum Complexity

We analyze relationships between quantum computation and a family of generalizations of the Jones polynomial. Extending recent work by Aharonov et al., we give efficient quantum circuits for implementing the unitary Jones-Wenzl representations of the braid group. We use these to provide new quantum algorithms for approximately evaluating a family of specializations of the HOMFLYPT two-variable polynomial of trace closures of braids. We also give algorithms for approximating the Jones polynomial of a general class of closures of braids at roots of unity. Next we provide a self-contained proof of a result of Freedman et al.\ that any quantum computation can be replaced by an additive approximation of the Jones polynomial, evaluated at almost any primitive root of unity. Our proof encodes two-qubit unitaries into the rectangular representation of the eight-strand braid group. We then give QCMA-complete and PSPACE-complete problems which are based on braids. We conclude with direct proofs that evaluating the Jones polynomial of the plat closure at most primitive roots of unity is a \#P-hard problem, while learning its most significant bit is PP-hard, circumventing the usual route through the Tutte polynomial and graph coloring.

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.

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

JONES POLYNOMIAL INVARIANTS

2006 ◽  
Vol 15 (03) ◽  
pp. 339-350 ◽  
Author(s):  
ANDREW FISH ◽  
EBRU KEYMAN
Keyword(s):  
Jones Polynomial ◽  
Polynomial Invariants ◽  
Virtual Links

The Jones polynomial is a well-defined invariant of virtual links. We observe the effect of a generalised mutation M of a link on the Jones polynomial. Using this, we describe a method for obtaining invariants of links which are also invariant under M. The Jones polynomial of welded links is not well-defined in ℤ[q1/4, q-1/4]. Taking M = Fo allows us to pass to a quotient of ℤ[q1/4, q-1/4] in which the Jones polynomial is well-defined. We get the same result for M = Fu, so in fact, the Jones polynomial in this ring defines a fused isotopy invariant. We show it is non-trivial and compute it for links with one or two components.

scholarly journals Virtual Khovanov homology using cobordisms

2014 ◽  
Vol 23 (09) ◽  
pp. 1450046 ◽  
Author(s):  
Daniel Tubbenhauer
Keyword(s):  
Computer Program ◽  
Jones Polynomial ◽  
Virtual Link ◽  
Link Homology ◽  
Direct Extension ◽  
Virtual Links ◽  
Local Properties ◽  
Characteristic Two ◽  
Topological Construction

We extend Bar-Natan's cobordism-based categorification of the Jones polynomial to virtual links. Our topological complex allows a direct extension of the classical Khovanov complex (h = t = 0), the variant of Lee (h = 0, t = 1) and other classical link homologies. We show that our construction allows, over rings of characteristic two, extensions with no classical analogon, e.g. Bar-Natan's ℤ/2-link homology can be extended in two non-equivalent ways. Our construction is computable in the sense that one can write a computer program to perform calculations, e.g. we have written a MATHEMATICA-based program. Moreover, we give a classification of all unoriented TQFTs which can be used to define virtual link homologies from our topological construction. Furthermore, we prove that our extension is combinatorial and has semi-local properties. We use the semi-local properties to prove an application, i.e. we give a discussion of Lee's degeneration of virtual homology.

scholarly journals Knot theory and matrix integrals

2018 ◽  
Author(s):  
Jeremie Bouttier
Keyword(s):  
Knot Theory ◽  
Virtual Knots ◽  
Quartic Potential ◽  
Link Diagrams ◽  
Large Size ◽  
Virtual Links ◽  
Matrix Integrals ◽  
Higher Genus ◽  
Enumeration Problems ◽  
Alternating Links

This article considers some enumeration problems in knot theory, with a focus on the application of matrix integral techniques. It first reviews the basic definitions of knot theory, paying special attention to links and tangles, especially 2-tangles, before discussing virtual knots and coloured links as well as the bare matrix model that describes coloured link diagrams. It shows how the large size limit of matrix integrals with quartic potential may be used to count alternating links and tangles. The removal of redundancies amounts to renormalization of the potential. This extends into two directions: first, higher genus and the counting of ‘virtual’ links and tangles, and second, the counting of ‘coloured’ alternating links and tangles. The article analyses the asymptotic behaviour of the number of tangles as the number of crossings goes to infinity

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.

A Tutte Polynomial for Coloured Graphs

1999 ◽  
Vol 8 (1-2) ◽  
pp. 45-93 ◽  
Author(s):  
BÉLA BOLLOBÁS ◽  
OLIVER RIORDAN
Keyword(s):  
Sufficient Conditions ◽  
Jones Polynomial ◽  
Tutte Polynomial ◽  
Graph Invariant ◽  
Link Invariant ◽  
Random Cluster Model ◽  
Tree Expansion ◽  
Function Formulation ◽  
The One ◽  
Necessary And Sufficient

We define a polynomial W on graphs with colours on the edges, by generalizing the spanning tree expansion of the Tutte polynomial as far as possible: we give necessary and sufficient conditions on the edge weights for this expansion not to depend on the order used. We give a contraction-deletion formula for W analogous to that for the Tutte polynomial, and show that any coloured graph invariant satisfying such a formula can be obtained from W. In particular, we show that generalizations of the Tutte polynomial obtained from its rank generating function formulation, or from a random cluster model, can be obtained from W. Finally, we find the most general conditions under which W gives rise to a link invariant, and give as examples the one-variable Jones polynomial, and an invariant taking values in ℤ/22ℤ.

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.

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