Instanton knot invariants with rational holonomy parameters and an application for torus knot groups

TL;DR

This paper develops new instanton knot invariants with rational holonomy parameters, extending existing theories and providing insights into torus knot groups and their representations, with implications for the Slice-Ribbon conjecture.

Contribution

It generalizes Daemi-Scaduto's equivariant singular instanton Floer theory to include rational holonomy parameters for torus knots.

Findings

Irreducible singular instanton homology of torus knots determined for most rational parameters.

Provides evidence supporting the Slice-Ribbon conjecture for torus knots.

Extends the understanding of knot group representations using instanton invariants.

Abstract

There are several knot invariants in the literature that are defined using singular instantons. Such invariants provide strong tools to study the knot group and give topological applications. For instance, it gives powerful tools to study the topology of knots in terms of representations of fundamental groups. In particular, it is shown that any traceless representation of the torus knot group can be extended to any concordance from the torus knot to another knot. Daemi and Scaduto proposed a generalization that is related to a version of the Slice-Ribbon conjecture to torus knots. The results of this paper provide further evidence towards the positive answer to this question. The method is a generalization of Daemi-Scaduto's equivariant singular instanton Floer theory following Echeverria's earlier work. Moreover, the irreducible singular instanton homology of torus knots for all but finitely many rational holonomy parameters are determined as $\mathbb{Z}/4$-graded abelian groups.