Genus bounds from quantum invariants
Moduli and Friends seminar
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Monday 9 May 2022

Daniel López Neumann (Indiana University, Bloomington)Genus bounds from quantum invariants

It is a well-known theorem that the degree of the Alexander polynomial of a knot gives a lower bound to the Seifert genus, that is, the minimal genus of a surface embedded in the three-sphere bounding the given knot. However, the so called quantum invariants of knots, such as the Jones polynomial, are usually unrelated to the Seifert genus.

In this talk, we will explain a general procedure to build knot polynomials that satisfy a genus bound. The main idea is to use Turaev's G-graded extension of the theory of quantum invariants applied to the "twisted Drinfeld double" of a Hopf algebra. Our genus bound recovers those from twisted Alexander polynomials (due to Friedl-Kim) and produce new bounds for "non-semisimple" quantum invariants, such as the Akutsu-Deguchi-Ohtsuki (ADO) invariants. This is work in progress with Roland van der Veen.

* Eastern European Time, i.e. UTC+2 in winter and UTC+3 in summer.