Alexander polynomial, Dijkgraaf-Witten invariant, and Seifert fibred surgery

TL;DR

This paper links the Dijkgraaf-Witten invariant and Alexander polynomial to derive constraints on knot surgeries resulting in small Seifert 3-manifolds, providing new insights into 3-manifold topology.

Contribution

It introduces a novel application of the Dijkgraaf-Witten invariant to relate Alexander polynomial data to Seifert fibered surgeries on knots.

Findings

Constraints on surgery parameters derived from Alexander polynomial

Connections between Dijkgraaf-Witten invariant and Seifert fibered manifolds

New methods for analyzing knot surgeries

Abstract

We apply Dijkgraaf-Witten invariant over an semiproduct of abelian groups to show that, if the $k/\ell$-surgery along a knot $K$ results in a small Seifert 3-manifold with multiplicities $a_1,a_2,a_3$, then many constraints on $k,a_1,a_2,a_3$ can be read off from the Alexander polynomial of $K$.