Parabolic Dijkgraaf-Witten invariants of links in the $3$-sphere

TL;DR

This paper introduces a new link invariant called the parabolic Dijkgraaf-Witten invariant, generalizing previous invariants, with computations for links related to lens spaces and a diagram-based calculation method.

Contribution

It defines the parabolic Dijkgraaf-Witten invariant, extending prior invariants, and provides computational techniques using link diagrams.

Findings

Computed the invariant for links with double branched coverings homeomorphic to lens spaces.

Established a diagram-based procedure for partial invariant computation.

Demonstrated the invariant's generalization of the reduced DW invariant.

Abstract

We define a new invariant of links in the $3$-sphere and call it the parabolic Dijkgraaf-Witten (DW) invariant. This invariant is a generalization of the reduced DW invariant derived by Karuo. In this paper, we compute the invariant of several links over which double branched coverings are homeomorphic to the lens spaces. Moreover, we introduce a procedure for computing partial information of the parabolic DW invariant using only link diagrams.