On the fundamental 3-classes of knot group representations

TL;DR

This paper explores the fundamental 3-classes of knot groups, establishing a connection between group homology and quandle homology, and provides an algorithm for hyperbolic knots.

Contribution

It introduces a diagrammatic approach to push-forwards of knot group representations and presents an algorithm to compute the fundamental 3-class of hyperbolic knots.

Findings

Bridge between relative group homology and quandle homology

Diagrammatic descriptions of push-forwards for link-group representations

Algorithm for algebraic description of the fundamental 3-class in hyperbolic knots

Abstract

We discuss the fundamental (relative) 3-classes of knots (or hyperbolic links), and provide diagrammatic descriptions of the push-forwards with respect to every link-group representation. The point is an observation of a bridge between the relative group homology and quandle homology from the viewpoints of Inoue--Kabaya map \cite{IK}. Furthermore, we give an algorithm to algebraically describe the fundamental 3-class of any hyperbolic knot.