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.

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

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.

Jones polynomials and classical conjectures in knot theory. II

1987 ◽  
Vol 102 (2) ◽  
pp. 317-318 ◽  
Author(s):  
Kunio Murasugi
Keyword(s):  
Knot Theory ◽  
Jones Polynomial ◽  
Jones Polynomials ◽  
Alternating Link

Let L be an alternating link and be its reduced (or proper) alternating diagram. Let w() denote the writhe of [3], i.e. the number of positive crossings minus the number of negative crossings. Let VL(t) be the Jones polynomial of L [2]. Let dmaxVL(t) and dminVL(t) denote the maximal and minimal degrees of VL(t), respectively. Furthermore, let σ(L) be the signature of L [5].

scholarly journals Zeros of the Jones Polynomial are Dense in the Complex Plane

10.37236/366 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Xian'an Jin ◽  
Fuji Zhang ◽  
Fengming Dong ◽  
Eng Guan Tay
Keyword(s):  
Complex Plane ◽  
Jones Polynomial ◽  
Tutte Polynomial ◽  
Signed Graph ◽  
Labeled Graph ◽  
Jones Polynomials

In this paper, we present a formula for computing the Tutte polynomial of the signed graph formed from a labeled graph by edge replacements in terms of the chain polynomial of the labeled graph. Then we define a family of 'ring of tangles' links and consider zeros of their Jones polynomials. By applying the formula obtained, Beraha-Kahane-Weiss's theorem and Sokal's lemma, we prove that zeros of Jones polynomials of (pretzel) links are dense in the whole complex plane.

scholarly journals Bounds on the Complex Zeros of (Di)Chromatic Polynomials and Potts-Model Partition Functions

2001 ◽  
Vol 10 (1) ◽  
pp. 41-77 ◽  
Author(s):  
ALAN D. SOKAL
Keyword(s):  
Partition Function ◽  
Simple Proof ◽  
Potts Model ◽  
Maximum Degree ◽  
Partition Functions ◽  
Polymer Model ◽  
Chromatic Polynomials ◽  
Reliability Polynomial ◽  
Complex Zeros ◽  
Special Case

We show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree [les ] r, the zeros (real or complex) of the chromatic polynomial PG(q) lie in the disc [mid ]q[mid ] < C(r). Furthermore, C(r) [les ] 7.963907r. This result is a corollary of a more general result on the zeros of the Potts-model partition function ZG(q, {ve}) in the complex antiferromagnetic regime [mid ]1 + ve[mid ] [les ] 1. The proof is based on a transformation of the Whitney–Tutte–Fortuin–Kasteleyn representation of ZG(q, {ve}) to a polymer gas, followed by verification of the Dobrushin–Kotecký–Preiss condition for nonvanishing of a polymer-model partition function. We also show that, for all loopless graphs G of second-largest degree [les ] r, the zeros of PG(q) lie in the disc [mid ]q[mid ] < C(r) + 1. Along the way, I give a simple proof of a generalized (multivariate) Brown–Colbourn conjecture on the zeros of the reliability polynomial for the special case of series-parallel graphs.

scholarly journals Character decomposition of Potts model partition functions, II: Toroidal geometry

2006 ◽  
Vol 750 (3) ◽  
pp. 229-249 ◽  
Author(s):  
Jean-François Richard ◽  
Jesper Lykke Jacobsen
Keyword(s):  
Potts Model ◽  
Partition Functions ◽  
Toroidal Geometry

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.

THE COLORED JONES POLYNOMIALS OF DOUBLES OF KNOTS

2008 ◽  
Vol 17 (08) ◽  
pp. 925-937
Author(s):  
TOSHIFUMI TANAKA
Keyword(s):  
Jones Polynomial ◽  
Colored Jones Polynomial ◽  
Volume Conjecture ◽  
Jones Polynomials ◽  
Skein Theory ◽  
Colored Jones Polynomials

We give formulas for the N-colored Jones polynomials of doubles of knots by using skein theory. As a corollary, we show that if the volume conjecture for untwisted positive (or negative) doubles of knots is true, then the colored Jones polynomial detects the unknot.

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