On the Seifert fibered space link group

TL;DR

This paper introduces new diagrammatic tools for links in Seifert fibered spaces, enabling computation of their fundamental groups, homology, and twisted Alexander polynomials, advancing understanding of their topological properties.

Contribution

It develops generalized arrow diagrams and Reidemeister moves for links in Seifert fibered spaces, providing a method to compute their fundamental groups and related invariants.

Findings

Presented a presentation of the link complement's fundamental group.

Computed the first homology group of the link complement.

Calculated twisted Alexander polynomials for links in Seifert fibered spaces.

Abstract

We introduce generalized arrow diagrams and generalized Reidemeister moves for diagrams of links in Seifert fibered spaces. We give a presentation of the fundamental group of the link complement. As a corollary we are able to compute the first homology group of the complement and the twisted Alexander polynomials of the link.