POSITIVITY OF KNOT POLYNOMIALS ON POSITIVE LINKS

On Polynomials and Surfaces of Variously Positive Links

It is known that the minimal degree of the Jones polynomial of a positive knot is equal to its genus, and the minimal coefficient is 1, with a similar relation for links. We extend this result to almost positive links and partly identify the 3 following coefficients for special types of positive links. We also give counterexamples to the Jones polynomial-ribbon genus conjectures for a quasiposi...

Knot Polynomials and Knot Homologies

This is an expository paper discussing some parallels between the Khovanov and knot Floer homologies. We describe the formal similarities between the theories, and give some examples which illustrate a somewhat mysterious correspondence between them.

Positivity Of Trigonometri Polynomials

Knot polynomials via one parameter knot theory

We construct new knot polynomials. Let V be the standard solid torus in 3-space and let pr be its standard projection onto an annulus. LetM be the space of all smooth oriented knots in V such that the restriction of pr is an immersion (e.g. regular diagrams of a classical knot in the complement of its meridian). There is a canonical one dimensional homology class for each connected component of...

Knot Colouring Polynomials

This article introduces a natural extension of colouring numbers of knots, called colouring polynomials, and studies their relationship to Yang-Baxter invariants and quandle 2-cocycle invariants. For a knot K in the 3-sphere let πK be the fundamental group of the knot complement S 3 rK, and let mK , lK ∈ πK be a meridian-longitude pair. Given a finite group G and an element x ∈ G we consider th...