Zeros of Jones Polynomials for Families of Knots and Links

The Jones polynomial $V_{L}(t)$ for an oriented link $L$ is a one-variable Laurent polynomial link invariant discovered by Jones. For any integer $n\ge 3$, we show that: (1) the difference of Jones polynomials for two oriented links which are $C_{n}$-equivalent is divisible by $\left(t-1\right)^{n}\left(t^{2}+t+1\right)\left(t^{2}+1\right)$, and (2) there exists a pair of two oriented knots which are $C_{n}$-equivalent such that the difference of the Jones polynomials for the...

Using computer calculations and working with representatives of pretzel tangles we established general adequacy criteria for different classes of knots and links. Based on adequate graphs obtained from all Kauffman states of an alternating link we defined a new numerical invariant: adequacy number, and computed adequacy polynomial which is the invariant of alternating link families. Adequacy polynomial distinguishes (up to mutation) all families of alternating knots and links...

We identify all hyperbolic knots whose complements are in the census of orientable one-cusped hyperbolic manifolds with eight ideal tetrahedra. We also compute their Jones polynomials.

Kalfagianni and Lee found two-sided bounds for the crosscap number of an alternating link in terms of certain coefficients of the Jones polynomial. We show here that we can find similar two-sided bounds for the crosscap number of Conway sums of strongly alternating tangles. Then we find families of links for which these coefficients of the Jones polynomial and the crosscap number grow independently. These families will enable us to show that neither linear bound generalizes f...

Recently, Kashaev and the first author defined a sequence $V_n$ of 2-variable knot polynomials with integer coefficients, coming from the $R$-matrix of a rank 2 Nichols algebra, the first polynomial been identified with the Links--Gould polynomial. In this note we present the results of the computation of the $V_n$ polynomials for $n=1,2,3,4$ and discover applications and emerging patterns, including unexpected Conway mutations that seem undetected by the $V_n$-polynomials as...

A table of the families of alternating knots formed by conways is presented. The Conway's function is shown with the use of linear algebra in terms of natural numbers, called conways, that represent the number of crossings along a direction, as it was used by J. Conway for the classification of knots. Colored figures and tangles show the parts of the knots or tangles with a definite handedness: all the colored parts of the knot family are associated to a particular orientatio...

The present paper is an introduction to a combinatorial theory arising as a natural generalisation of classical and virtual knot theory. There is a way to encode links by a class of `realisable' graphs. When passing to generic graphs with the same equivalence relations we get `graph-links'. On one hand graph-links generalise the notion of virtual link, on the other hand they do not feel link mutations. We define the Jones polynomial for graph-links and prove its invariance. W...

We find families of prime knot diagrams with arbitrary extreme coefficients in their Jones polynomials. Some graph theory is presented in connection with this problem, generalizing ideas by Yongju Bae and Morton and giving a positive answer to a question in their paper.

Knots and links are interpreted as homotopy classes of nanowords and nanophrases in an alphabet consisting of 4 letters. Similar results hold for curves on surfaces. We also discuss versions of the Jones link polynomial and the link quandles for nanophrases.

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 Bollob\'as 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 graph...