Knotted Strings in the plane

Neslihan Güğümcü, Technical University of Athens

Date and Time: 15/03/2019, 10:00

Place: B256

Abstract: Planar curves have been studied since the time of Gauss. Gauss was one of the first to notice that they can be handled combinatorially by codes (named as Guass codes) that are strings of labels encoding self-intersections. Whitney classified all immersed curves up to a topological relation called regular homotopy by using the winding number of immersion maps. In the first half of the 20th century Reidemeister showed that classical knot theory is equivalent to the study of immersed curves in the plane, whose self-intersections are endowed with a combinatorial structure, with an under/over-data. With this extra structure, regular homotopy needs to transforms into a richer equivalence relation generated by Reidemeister moves. Since then knot theory is a classical subject of topology, bringing us many interesting questions relating to combinatorial topology.

In this talk, we will talk about knotoids (introduced by Turaev) that provide us a new diagrammatic theory that is an extension of classical knot theory. Problem of classifying knotoids lies at the center of the theory of knotoids. We will construct a Laurent polynomial with integer coefficients for knotoids called the affine index polynomial and we will show how it contributes to the classification problem.