The Kauman Polynomial and Trivalent Graphs

2018

We generalize some of the congruences in [20] to periodic knotted trivalent graphs. As an application, a criterion derived from one of these congruences is used to obstruct periodicity of links of few crossings for the odd primes p = 3, 5, 7, and 11. Moreover, we derive a new criterion of periodic links. In particular, we give a sufficient condition for the period to divide the crossing number. This gives some progress toward solving the well-known conjecture that the period divides the crossing number in the case of alternating links.

The Bollobas-Riordan polynomial [Math. Ann. 323, 81 (2002)] is a universal polynomial invariant for ribbon graphs. We find an extension of this polynomial for a particular family of graphs called rank 3 weakly-colored stranded graphs. These graphs live in a 3D space and appear as the gluing stranded vertices with stranded edges according to a definite rule (ordinary graphs and ribbon graphs can be understood in terms of stranded graphs as well). They also possess a color structure in a specific sense [Gurau, Commun. Math. Phys. 304, 69 (2011)]. The polynomial constructed is a seven indeterminate polynomial invariant of these graphs which responds to a similar contraction/deletion recurrence relation as obeyed by the Tutte and Bollobas-Riordan polynomials. It is however new due to the particular cellular structure of the graphs on which it relies. The present polynomial encodes therefore additional data that neither the Tutte nor the Bollobas-Riordan polynomials can capture for the t...

1985

The purpose of this note is to announce a new isotopy invariant of oriented links of tamely embedded circles in 3-space. We represent links by plane projections, using the customary conventions that the image of the link is a union of transversely intersecting immersed curves, each provided with an orientation, and undercrossings are indicated by broken lines. Following Conway [6], we use the symbols L+, Lo, L_ to denote links having plane projections which agree except in a small disk, and inside that disk are represented by the pictures of Figure 1. Conway showed that the one-variable Alexander polynomials of L+, Lo, L_ (when suitably normalized) satisfy the relation

Bulletin of the American Mathematical Society, 1985

The purpose of this note is to announce a new isotopy invariant of oriented links of tamely embedded circles in 3-space. We represent links by plane projections, using the customary conventions that the image of the link is a union of transversely intersecting immersed curves, each provided with an orientation, and undercrossings are indicated by broken lines. Following Conway [6], we use the symbols L+, Lo, L_ to denote links having plane projections which agree except in a small disk, and inside that disk are represented by the pictures of Figure 1. Conway showed that the one-variable Alexander polynomials of L+, Lo, L_ (when suitably normalized) satisfy the relation

Involve, a Journal of Mathematics, 2014

François Jaeger presented the two-variable Kauffman polynomial of an unoriented link L as a weighted sum of HOMFLY-PT polynomials of oriented links associated with L. Murakami, Ohtsuki and Yamada (MOY) used planar graphs and a recursive evaluation of these graphs to construct a state model for the sl(n)link invariant (a one-variable specialization of the HOMFLY-PT polynomial). We apply the MOY framework to Jaeger's work, and construct a state summation model for the SO(2n) Kauffman polynomial.

2019

This article provides an overview of relative strengths of polynomial invariants of knots and links, such as the Alexander, Jones, Homflypt, and Kaufman two-variable polynomial, Khovanov homology, factorizability of the polynomials, and knot primeness detection.

2020

A knot is an embedding of a circle S 1 into the three-dimensional sphere S 3 . An ncomponent link is an embedding of n disjoint circles ∏ n i=1 S 1 into S 3 . The main objective of knot theory is to classify knots and links up to natural deformations called isotopies. While there is no simple algorithm that helps decide whether two given knots (or links) are equivalent, various topological invariants have been developed to help distinguish between non-equivalent knots and links. The Alexander polynomial ∆ L (t) is one of the oldest such tools. It was originally defined from the Seifert surface of the knot. A simpler recursive definition of this invariant has been introduced later using Conway skein relations on the link diagram. The aim of this study is to compute the Alexander polynomial and obtain an explicit formula for some families of alternating knots of braid index 3. This formula is used to prove that ∆ L (t) satisfies Fox's trapezoidal conjecture. This conjecture states that the coefficients of the Alexander polynomial of an alternating knot are trapezoidal. In other words, these coefficients increase, stabilize then decrease in a symmetrical way. The main tool in this study is the Burau representation of the braid group.

Contemporary mathematics, 2017

2011

This article provides an overview of relative strengths of polynomial invariants of knots and links, such as the Alexander, Jones, Homflypt, and Kaufman two-variable polynomial, Khovanov homology, factorizability of the polynomials, and knot primeness detection.

Asian journal of natural and applied sciences, 2013

In this paper, we give some specializations and evaluations of the Tutte polynomial of a family of positive-signed connected planar graphs. First of all, we give the general form of the Tutte polynomial of the family of graphs using directly the deletion-contraction definition of the Tutte polynomial. Then, we give general formulas of Jones polynomials of very interesting families of alternating knots and links that correspond to these planar graphs; we actually specialize the Tutte polynomial to the Jones polynomial with the change of variables, and and with some factor of . In case of twocomponent links, we get two different formulas of the Jones polynomial, one when both the links are oriented either in clockwise or counterclockwise direction and another one when one component is oriented clockwise and the second counterclockwise. Moreover, we give general forms of the flow, reliability, and chromatic polynomials of these graphs. The reason to study flow polynomial is that it giv...