Simplicial Complexes on Seifert Surfaces of Links

Abstract

In this thesis project, we study various properties related to two well-known simplicial complexes on the Seifert surfaces of links - the incompressible complex and the Kakimizu complex (named after Osamu Kakimizu). We first look at Kakimizu’s original paper from 1992, in which he defines these complexes and describes a metric coming from the cyclic covering space of the knot complement, which is the same as the graph metric on the complexes. Then we study various properties of the Kakimizu complex, like connectedness, contractibility, local infiniteness and its coarse geometry. Finally, we try to extend some of these properties to the incompressible complex, and find a restriction on the types of knots that can have Kakimizu complex homeomorphic to the real line.