Month: November 2020

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 Gordon-Litherland pairing for knots and links in thickened surfaces

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Hans Boden (McMaster University)

We introduce the Gordon-Litherland pairing for knots and links in thickened surfaces that bound unoriented spanning surfaces. Using the GL pairing, we define signature and determinant invariants for such links. We relate the invariants to those derived from the Tait graph and Goeritz matrices. These invariants depend only on the $S^*$ equivalence class of the spanning surface, and the determinants give a simple criterion to check if the knot or link is minimal genus. This is joint work with M. Chrisman and H. Karimi. In further joint work with H. Karimi, we apply the GL pairing to give a topological characterization of alternating links in thickened surfaces, extending the results of J. Greene and J. Howie. 

Slides for talk:

GL-pairing-talk

 

Two-bridge knots and their crosscap numbers

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Cornelia Van Cott (University of San Francisco)

Begin with two knots K and J. Simon conjectured that if the knot group of K surjects onto that of J, then the genera of the orientable surfaces that the two knots bound are constrained. Specifically, he conjectured g(K) \geq g(J), where g(K) denotes the genus of K. This conjecture has been proved for alternating knots and can be strengthened to an even stronger result in the case of two-bridge knots. In this talk, we consider the same sorts of questions, but in the world of nonorientable surfaces. We focus on two-bridge knots and find relationships among their crosscap numbers. This is joint work with Jim Hoste and Pat Shanahan.

References:

Hoste, J., Shanahan, P., Van Cott, C., “Crosscap number and the partial order on two-bridge knots”, https://arxiv.org/abs/2010.05009

Further Reading:

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