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

Hayato Imori

Algebraic & Geometric Topology 24 (2024) 5045–5122
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, the topology of knots in terms of representations of fundamental groups. In particular, it has been 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 for torus knots. Our results provide further evidence towards the positive answer to this question. We use a generalization of Daemi and Scaduto’s equivariant singular instanton Floer theory following Echeverria’s earlier work. We also determine the irreducible singular instanton homology of torus knots for all but finitely many rational holonomy parameters as 4–graded abelian groups.

Keywords
instanton Floer homology, singular instanton knot homology, Levine–Tristram signature, slice-ribbon conjecture
Mathematical Subject Classification
Primary: 57R58
Secondary: 57K18
References
Publication
Received: 15 January 2023
Revised: 13 November 2023
Accepted: 5 December 2023
Published: 27 December 2024
Authors
Hayato Imori
Department of Mathematical Sciences
Korea Advanced Institute of Science and Technology
Daejeon
South Korea

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