[Paper Review] Quandle cocycle invariants of links using Mochizuki's 3-cocycles and Dijkgraaf-Witten invariants of 3-manifolds

Takefumi Nosaka|arXiv (Cornell University)|Mar 20, 2011

Geometric and Algebraic Topology13 references3 citations

TL;DR

This paper establishes a connection between quandle cocycle invariants of links and Dijkgraaf-Witten invariants of 3-manifolds via Mochizuki's 3-cocycles on Alexander quandles over finite fields. It shows that non-2-cocycle 3-cocycles arise from group 3-cocycles of a meta-abelian group, and demonstrates that the quandle cocycle invariant is equivalent to the ℤ-equivariant part of the Dijkgraaf-Witten invariant for cyclic branched covers over links in S³, enabling computation of Massey triple products through this framework.

ABSTRACT

T. Mochizuki determined all 3-cocycles of the third quandle cohomologies of Alexander quandles on finite fields. We show that all the 3-cocycles, except those of 2-cocycle forms, are derived from group 3-cocycles of a meta-abelian group. Further, the quandle cocycle invariant of a link using Mochizuki's 3-cocycle is equivalent to a $\Z$-equivariant part of the Dijkgraaf-Witten invariant of a cyclic covering of $S^3$ branched over the link using the group. We compute some Massey triple products via the former invariant.

Motivation & Objective

• To classify all 3-cocycles in the third quandle cohomology of Alexander quandles over finite fields.
• To determine the algebraic origin of non-2-cocycle 3-cocycles in terms of group cohomology.
• To establish a correspondence between quandle cocycle invariants of links and Dijkgraaf-Witten invariants of cyclic branched covers of S³.
• To apply this correspondence to compute Massey triple products using quandle cocycle invariants.

Proposed method

• Utilize Mochizuki's classification of 3-cocycles in the third cohomology of Alexander quandles over finite fields.
• Show that all 3-cocycles, except those of 2-cocycle type, are induced by group 3-cocycles of a meta-abelian group.
• Construct a cyclic covering of S³ branched over a link and relate the Dijkgraaf-Witten invariant of the resulting 3-manifold to the quandle cocycle invariant.
• Identify the ℤ-equivariant part of the Dijkgraaf-Witten invariant as equivalent to the quandle cocycle invariant.
• Use the equivalence to compute Massey triple products via the quandle cocycle invariant.
• Leverage the algebraic structure of the meta-abelian group to derive cohomological properties of the invariants.

Experimental results

Research questions

• RQ1 What is the group-theoretic origin of 3-cocycles in the quandle cohomology of Alexander quandles over finite fields, excluding 2-cocycle forms?
• RQ2 How are quandle cocycle invariants of links related to Dijkgraaf-Witten invariants of 3-manifolds obtained as cyclic branched covers of S³ over links?
• RQ3 Can the ℤ-equivariant part of the Dijkgraaf-Witten invariant be identified with the quandle cocycle invariant in this context?
• RQ4 To what extent can Massey triple products be computed using quandle cocycle invariants via this correspondence?
• RQ5 What role does the meta-abelian group structure play in the construction of these 3-cocycles?

Key findings

• All 3-cocycles in the third quandle cohomology of Alexander quandles over finite fields, except those of 2-cocycle form, arise from group 3-cocycles of a meta-abelian group.
• The quandle cocycle invariant of a link using Mochizuki's 3-cocycle is equivalent to the ℤ-equivariant part of the Dijkgraaf-Witten invariant of the cyclic branched cover of S³ over the link.
• The correspondence allows the computation of Massey triple products through the quandle cocycle invariant framework.
• The construction reveals a deep algebraic link between quandle cohomology and group cohomology in the context of 3-manifold invariants.
• The equivalence holds specifically for cyclic branched covers, highlighting the role of symmetry in the invariant construction.
• The method provides a new computational tool for Massey triple products using quandle-theoretic invariants.