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

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.

arXiv comment66 pages, 7 figures, Theorem 1.7, 1.8, and 1.10 are improved

References in corpus (8)

• Floer homology and knot complements
• Knot homology groups from instantons
• Filtered instanton Floer homology and the homology cobordism group
• Chern-Simons functional, singular instantons, and the four-dimensional clasp number
• Chern-Simons Functional and the Homology Cobordism Group
• A Generalization of the Tristram-Levine Knot Signatures as a Singular Furuta-Ohta Invariant for Tori
• Equivariant aspects of singular instanton Floer homology
• Link homology and equivariant gauge theory

Cited by in corpus (1)

• Instantons, special cycles, and knot concordance