Ascending number and Conway polynomial

For a knot, the ascending number is the minimum number of crossing changes which are needed to obtain an ascending diagram. We study the ascending number of a knot by analyzing the Conway polynomial. In this paper, we give a sharper lower bound of the ascending number of a knot and newly determine the ascending number for 26 prime knots up to 10 crossings.

The Conway polynomial and amphicheiral knots

According to work of Hartley and Kawauchi in 1979 and 1980, the Conway Polynomial of all negative amphicheiral knots and strongly positive amphicheiral knots factors as [Formula: see text] for some [Formula: see text]. Moreover, a 2012 example due to Ermotti, Hongler and Weber shows that this is not true for general amphicheiral knots. On the other hand, in 2006, the first author made a conjecture equivalent to saying that the Conway polynomial of all amphicheiral knots splits as [Formula: see text] in the ring [Formula: see text]. In this paper, we establish this conjecture for all periodically amphicheiral knots built from braids, where the period preserves the braid structure. We also give counterexamples to conjectures on the leading coefficient of the Conway polynomial of an amphicheiral knot due to Stoimenow.

Strongly quasipositive links with braid index 3 have positive Conway polynomial

Strongly quasipositive links are those links which can be seen as closures of positive braids in terms of band generators. In this paper, we give a necessary condition for a link with braid index 3 to be strongly quasipositive, by proving that in that case, it has positive Conway polynomial (that is, all its coefficients are non-negative). We also show that this result cannot be extended to a higher number of strands, as we provide a strongly quasipositive braid whose closure has non-positive Conway polynomial.

Alexander–Conway polynomial state model and link homology

This paper shows how the Formal Knot Theory state model for the Alexander–Conway polynomial is related to Knot Floer Homology. In particular, we prove a parity result about the states in this model that clarifies certain relationships of the model with Knot Floer Homology.

On the topology of the coefficients of the Alexander–Conway polynomials of knots

Using a special form of spanning surface for a knot, we give a formula for the coefficient of the [Formula: see text]-term of the Alexander–Conway polynomial in terms of the sum of determinants of the blocks of [Formula: see text] submatrices of the Seifert matrix, from which the topological meaning of the coefficient is revealed.

On the Links–Gould invariant and the square of the Alexander polynomial

This paper gives a connection between well-chosen reductions of the Links–Gould invariants of oriented links and powers of the Alexander–Conway polynomial. This connection is obtained by showing the representations of the braid groups we derive the specialized Links–Gould polynomials from can be seen as exterior powers of a direct sum of Burau representations.

HOMFLY Polynomials of Torus Links as Generalized Fibonacci Polynomials

The focus of this paper is to study the HOMFLY polynomial of $(2,n)$-torus link as a generalized Fibonacci polynomial. For this purpose, we first introduce a form of generalized Fibonacci and Lucas polynomials and provide their some fundamental properties. We define the HOMFLY polynomial of $ (2,n) $-torus link with a way similar to our generalized Fibonacci polynomials and provide its fundamental properties. We also show that the HOMFLY polynomial of $ (2,n) $-torus link can be obtained from its Alexander-Conway polynomial or the classical Fibonacci polynomial. We finally give the matrix representations and prove important identities, which are similar to the Fibonacci identities, for the our generalized Fibonacci polynomial and the HOMFLY polynomial of $ (2,n) $-torus link.