Alternating diagram

A volumish theorem for alternating virtual links

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Ilya Kofman (CUNY)

Dasbach and Lin proved a “volumish theorem” for alternating links. We prove the analogue for alternating link diagrams on surfaces, which provides bounds on the hyperbolic volume of a link in a thickened surface in terms of coefficients of its reduced Jones-Krushkal polynomial. Joint work with Abhijit Champanerkar.

References:

Champanerkar, Abhijit and Kofman, Ilya, “A volumish theorem for alternating virtual links”, https://arxiv.org/abs/2010.08499

The geometric content of Tait’s conjectures

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Thomas Kindred (University of Nebraska-Lincoln)

In 1898, Tait asserted several properties of alternating knot diagrams, which remained unproven until the discovery of the Jones polynomial in 1985. During that time, Fox asked, “What geometrically is an alternating knot?” By 1993, the Jones polynomial had led to proofs of all of Tait’s conjectures, but the geometric content of these new results remained mysterious. In 2017, Howie and Greene independently answered Fox’s question, and Greene used his characterization to give the first purely geometric proof of part of Tait’s conjectures. Recently, I used Greene and Howie’s characterizations, among other techniques, to give the first entirely geometric proof of Tait’s flyping conjecture (first proven in 1993 by Menasco and Thistlethwaite). I will describe these recent developments and sketch approaches to other parts of Tait’s conjectures, and related facts about tangles and adequate knots, which remain unproven by purely geometric means.

References:

Kindred, Thomas, “A geometric proof of the flyping theorem”, https://arxiv.org/abs/2008.06490

Slides for talk:

TKindred-CKVK_-9Nov2020

Unknotting with a single twist

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Samantha Allen (Dartmouth)

Ohyama showed that any knot can be unknotted by performing two full twists, each on a set of parallel strands. We consider the question of whether or not a given knot can be unknotted with a single full twist, and if so, what are the possible linking numbers associated to such a twist. It is observed that if a knot can be unknotted with a single twist, then some surgery on the knot bounds a rational homology ball. Using tools such as classical invariants and invariants arising from Heegaard Floer theory, we give obstructions for a knot to be unknotted with a single twist of a given linking number. In this talk, I will discuss some of these obstructions, their implications (especially for alternating knots), many examples, and some unanswered questions. This talk is based on joint work with Charles Livingston.

The Jones-Krushkal polynomial and minimal diagrams of surface links

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Homayun Karimi (McMaster University)

We prove a Kauffman-Murasugi-Thistlethwaite theorem for alternating
links in thickened surfaces. It implies that any reduced alternating diagram of a link
in a thickened surface has minimal crossing number, and any two reduced alternating
diagrams of the same link have the same writhe. This result is proved more generally
for link diagrams that are adequate, and the proof involves a two-variable generalization
of the Jones polynomial for surface links defined by Krushkal. The main result is
used to establish the first and second Tait conjectures for links in thickened surfaces
and for virtual links.  This is joint work with Hans Boden.