The Alexander and Markov theorems for links with symmetries

Speaker:

Alice Merz

Affiliation:

University of Lille

Date:

Tue, 05/11/2024 - 11:00 - 12:15

Location:

MPIM Lecture Hall [2]

Parent event:

Low-dimensional topology seminar [3]

The Alexander theorem (1923) and the Markov theorem (1936) are two classical results in knot theory that show respectively that every link can be represented as the closure of a braid and that braids that have the same closure are related by a finite number of simple operations, namely conjugation and (de-)stabilization.

In this talk we will construct an equivariant closure operator that takes in input two braids with a particular symmetry, called palindromic braids, and outputs a link that is preserved by an involution. Links with such symmetry are called strongly involutive, and when we restrict ourselves to knots they form a well-studied class of knots, called strongly invertible. We will hence give analogues of the Alexander and Markov theorems for the equivariant closure operator. In fact we will show that every strongly involutive link is the equivariant closure of two palindromic braids, drawing a parallel to the Alexander theorem. Moreover, we will see that any two pairs of palindromic braids yielding the same strongly involutive link are related by some operations akin to conjugation and (de-)stabilization.