Knots and links in dimensions 3 and 4

Chebyshev Presentation of Handle Sliding Relations in Skein Modules 

Skein modules were introduced by Józef H. Przytycki as generalisations of the Jones and HOMFLYPT polynomial link invariants in the 3-sphere to arbitrary 3-manifolds. The Kauffman bracket skein module (KBSM) is the most extensively studied of all. However, computing the KBSM of a 3-manifold is known to be notoriously hard, especially over the ring of Laurent polynomials. With the goal of finding a definite structure of the KBSM over this ring, several conjectures and theorems were stated over the years for KBSMs. We show that some of these conjectures, and even theorems, are not true. In this talk I will briefly discuss a counterexample to Marche’s generalisation of Witten’s conjecture. I will show that a theorem stated by Przytycki in 1999 about the KBSM of the connected sum of two handlebodies does not hold. I will also give the exact structure of the KBSM of of the connected sum of two solid tori and show that it is isomorphic to the KBSM of a genus two handlebody modulo some specific handle sliding relations. Moreover, these handle sliding relations can be written in terms of Chebyshev polynomials. Paper to appear (Joint with Thang Lê and Józef Przytycki) 

Link Floer Homology and Link Detection

Knot Floer homology is a powerful invariant of links due independently to Ozsváth-Szabó and J.Rasmussen. In this talk I will discuss links with the same knot Floer homology as (2,2n) cables of L-space knots. In particular, this analysis shows that knot Floer homology detects, say, the (2,8) cable of T(2,5). Corresponding results will be given for links with the same link Floer homology as (m,mn) cables of L-space knots. This is based on joint work with Subhankar Dey.

Paper: Cable Links, Annuli and Sutured Floer homology, with Subhankar Dey.  

The Jones Polynomial of a Fibered Positive Link

The maximum degree of the Jones polynomial of a fibered positive knot is at most four times the minimum degree, and we have a similar result for links. This theorem allows us to complete the positivity classification of knots up to 12 crossings. We introduce the viewer to Balanced and Burdened link diagrams, new kinds of diagrams we constructed in order to prove our theorem. We explore properties of these diagrams, and the related concepts and strategies involved in the proof. This talk is based on our paper published to the arXiv earlier this year. 

Paper: A new condition on the Jones polynomial of a fibered positive link 

Negative amphichiral knots and the half-Conway polynomial

This talk is an introduction to the half-Conway polynomial for strongly negative amphichiral knots, which is an equivariant analog of the Conway polynomial. As applications, I will talk about how studying the half-Conway polynomial has led to finding the first examples of non-slice amphichiral knots of determinant one, and that strongly negative amphichiral knots with trivial Alexander polynomial are equivariantly topologically slice. This talk is based on joint work with Keegan Boyle.

Papers:

1. Negative amphichiral knots and the half-Conway polynomial, with Keegan Boyle

2. Equivariant topological slice disks and negative amphichiral knots, with Keegan Boyle. 

Relating unknotting and untwisting operations

The untwisting number of a knot K is the minimum number of (null-homologous) twists required to convert K to the unknot. Such a twist can be viewed as a generalized crossing change so that the untwisting number is a direct generalization of unknotting number. I'll discuss relationships between these invariants (and other generalizations like the surgery description number) and discuss connections between these invariants and 4-dimensional topology.