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

TL;DR

This work defines a parabolic Dijkgraaf-Witten invariant for links in $S^3$ with target group $\mathrm{SL}_2(\mathbb{F}_q)$ and establishes its relationship to Karuo’s reduced DW invariant. It provides a definition via parabolic representations and $p$-fold branched covers, proves invariance, and, in the case where $B_L^2$ is a lens space, gives a decomposition that reduces computation to tractable pieces. A key contribution is a diagrammatic route to partial information and a bridge to quandle cocycle invariants, using the Bloch group to construct explicit quandle 3-cocycles $\psi_r$ that connect $DW_q(L)$ with Bloch-group evaluations. The paper also delivers explicit computations for families of links (e.g., $(2,2m)$-torus links and $2m$-twist knots) and demonstrates how the parabolic invariant lifts the reduced DW invariant in the generic $q$ regime, enabling practical, diagram-based calculations and deeper connections to quandle theory.

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.

Figures (6)

• Figure 1: The links $L_1, L_2$, and $L_3$.
• Figure 2: Bands $\mathcal{D}$ with $L_2$ and $L_3$.
• Figure 3: $(2,2m)$-torus link $\mathcal{T}_m$ and $2m$-twist knot $\mathcal{K}_m$.
• Figure 4: Arcs of $\mathcal{T}_m$.
• Figure 5: Coloring conditions at each crossing and around arcs.

Theorems & Definitions (16)

• Definition 2.1
• Remark 2.2
• Theorem 2.3
• proof
• Lemma 2.4
• proof
• Corollary 2.5
• proof
• Proposition 3.1
• Theorem 3.2