Obstructions to fibredness from quantum groups at roots of unity

YMSC Topology Seminar

Organizers：

陈伟彦、高鸿灏、黄意、林剑锋

Speaker：

Daniel Alonso Gabriel LOPEZ NEUMANN

University of Concepción

Time：

Thur., 16:00-17:00, Oct. 10, 2024

Venue：

B725, Shuangqing Complex Building

清华大学双清综合楼A座B725

Online：

Zoom Meeting ID: 405 416 0815

Passcode: 111111

Title：

Obstructions to fibredness from quantum groups at roots of unity

Abstract:

The quantum invariants of knots and 3-manifolds form a wide class of topological invariants defined through the theory of tensor categories, Hopf algebras, and quantum groups. These invariants have been mostly studied in the case of quantum groups at generic parameter (e.g. the Jones and HOMFLY polynomials) in which case their relation to the geometry of the knot complement is poorly understood, and is the subject of various conjectures.

In this talk, I'll show that a different situation happens if the parameter is non-generic, i.e., a root of unity: the knot polynomials that one obtains (which were originally defined by Akutsu-Deguchi-Ohtsuki) are related to the Seifert genus and fibredness, generalizing properties of the classical Alexander polynomial. I'll explain the ideas behind the fibredness criterion, this relies on a theorem of Giroux-Goodman that characterizes fiber surfaces in the three-sphere.