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

Hayato Imori

TL;DR

The work extends instanton knot invariants to general rational holonomy parameters, removing the traceless constraint in key extension results for torus knots. It builds a robust $ ext{S}$-complex framework with Frøyshov-type invariants for general holonomy, establishing a lifting/branched-cover correspondence and an absolute counting formula linking representation varieties to Tristram–Levine signatures. A central outcome is that for a knot $K$ concordant to a torus knot $T_{p,q}$, any $SU(2)$ representation of the torus knot group extends over the concordance complement, and the irreducible instanton knot homology of torus knots can be explicitly described for almost all rational holonomy parameters. The results provide evidence towards a gauge-theoretic formulation of slice-ribbon phenomena and offer new tools for analyzing knot concordance through fundamental-group representations and branched-cover techniques.

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.

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

TL;DR

The work extends instanton knot invariants to general rational holonomy parameters, removing the traceless constraint in key extension results for torus knots. It builds a robust -complex framework with Frøyshov-type invariants for general holonomy, establishing a lifting/branched-cover correspondence and an absolute counting formula linking representation varieties to Tristram–Levine signatures. A central outcome is that for a knot concordant to a torus knot , any representation of the torus knot group extends over the concordance complement, and the irreducible instanton knot homology of torus knots can be explicitly described for almost all rational holonomy parameters. The results provide evidence towards a gauge-theoretic formulation of slice-ribbon phenomena and offer new tools for analyzing knot concordance through fundamental-group representations and branched-cover techniques.

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 -graded abelian groups.

Paper Structure

This paper contains 31 sections, 65 theorems, 391 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Summary of Results
  4. Outline of the paper
  5. Backgrouds on singular instantons
  6. The space of singular connections
  7. The Chern-Simons functional
  8. The flip symmetry
  9. Holonomy perturbations
  10. The moduli space over the cylinder
  11. Compactness
  12. Cobordisms
  13. Orientation
  14. $\mathcal{S}$-complexs and Frøyshov type invariants
  15. A Review on $\mathcal{S}$-complexs and Frøyshov invariants
  16. ...and 16 more sections

Key Result

Theorem 1

(DS2 ) Let $S:T_{p,q}\rightarrow K$ be a given smooth concordance. Then any traceless $SU(2)$-representation of $\pi_{1}(S^{3}\setminus T_{p, q})$ extends over the concordance complement.

Figures (7)

  • Figure 1: Crossings of knot
  • Figure 2: Black dots represent lifts of the irreducible flat connection $\beta$ and red dots represent lift of the reducible flat connection $\theta_{\alpha}$.
  • Figure 3: Elements $\hat{\beta}_{0},\cdots, \hat{\beta}_{d-1}$ and their $\mathbb{Z}\times \mathbb{R}$-gradings.
  • Figure 4: The family of metric $G^{o}$
  • Figure 5: Paths on $(W^o, S^{o})$
  • ...and 2 more figures

Theorems & Definitions (144)

  • Theorem 1
  • Conjecture 2
  • Theorem 3
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • ...and 134 more