Seifert surfaces, Commutators and Vassiliev invariants

TL;DR

This paper explores how Vassiliev invariants serve as obstructions to constructing simple Seifert surfaces for knots, linking knot invariants with the algebraic structure of their complements.

Contribution

It establishes a connection between Vassiliev invariants and the lower central series of the fundamental group of Seifert surface complements, providing new insights into knot theory.

Findings

Vassiliev invariants obstruct the existence of simple Seifert surfaces.

The complement's fundamental group structure relates to knot invariants.

New criteria for Seifert surface complexity based on invariants.

Abstract

We show that the Vassiliev invariants of a knot K, are obstructions to finding a regular Seifert surface, S, whose complement looks "simple" (e.g. like the complement of a disc) to the lower central series of its fundamental group.