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Equivariant aspects of singular instanton Floer homology

Aliakbar Daemi, Christopher Scaduto

TL;DR

The paper develops a comprehensive S^1-equivariant refinement of singular instanton Floer theory for knots in integer homology 3-spheres. It builds an SU(2) singular connection framework with an S-complex backbone, yielding framed and equivariant knot invariants that recover and extend Kronheimer–Mrowka constructions, including I^\natural and I^#, and connect to Heegaard Floer-type structures through algebraic S-complexes. Central innovations include the framed knot complex tilde{C}(Y,K), the algebra of S-complexes with h and J_i invariants, and a connected-sum theorem that multiplicatively relates knot invariants under Y#Y'. The work also develops local coefficient systems and enriched S-complexes to define concordance invariants h_R and the Gamma-invariant, providing a refined, computable toolkit for knot concordance and representation-theoretic data in the instanton setting. Collectively, these results bridge singular instanton theory with equivariant, algebraic frameworks, yielding new tools for knot invariants, concordance bounds, and potential computational approaches via ADHM-type descriptions in spherical cases.

Abstract

We associate several invariants to a knot in an integer homology 3-sphere using $SU(2)$ singular instanton gauge theory. There is a space of framed singular connections for such a knot, equipped with a circle action and an equivariant Chern-Simons functional, and our constructions are morally derived from the associated equivariant Morse chain complexes. In particular, we construct a triad of groups analogous to the knot Floer homology package in Heegaard Floer homology, several Frøyshov-type invariants which are concordance invariants, and more. The behavior of our constructions under connected sums are determined. We recover most of Kronheimer and Mrowka's singular instanton homology constructions from our invariants. Finally, the ADHM description of the moduli space of instantons on the 4-sphere can be used to give a concrete characterization of the moduli spaces involved in the invariants of spherical knots, and we demonstrate this point in several examples.

Equivariant aspects of singular instanton Floer homology

TL;DR

The paper develops a comprehensive S^1-equivariant refinement of singular instanton Floer theory for knots in integer homology 3-spheres. It builds an SU(2) singular connection framework with an S-complex backbone, yielding framed and equivariant knot invariants that recover and extend Kronheimer–Mrowka constructions, including I^\natural and I^#, and connect to Heegaard Floer-type structures through algebraic S-complexes. Central innovations include the framed knot complex tilde{C}(Y,K), the algebra of S-complexes with h and J_i invariants, and a connected-sum theorem that multiplicatively relates knot invariants under Y#Y'. The work also develops local coefficient systems and enriched S-complexes to define concordance invariants h_R and the Gamma-invariant, providing a refined, computable toolkit for knot concordance and representation-theoretic data in the instanton setting. Collectively, these results bridge singular instanton theory with equivariant, algebraic frameworks, yielding new tools for knot invariants, concordance bounds, and potential computational approaches via ADHM-type descriptions in spherical cases.

Abstract

We associate several invariants to a knot in an integer homology 3-sphere using singular instanton gauge theory. There is a space of framed singular connections for such a knot, equipped with a circle action and an equivariant Chern-Simons functional, and our constructions are morally derived from the associated equivariant Morse chain complexes. In particular, we construct a triad of groups analogous to the knot Floer homology package in Heegaard Floer homology, several Frøyshov-type invariants which are concordance invariants, and more. The behavior of our constructions under connected sums are determined. We recover most of Kronheimer and Mrowka's singular instanton homology constructions from our invariants. Finally, the ADHM description of the moduli space of instantons on the 4-sphere can be used to give a concrete characterization of the moduli spaces involved in the invariants of spherical knots, and we demonstrate this point in several examples.

Paper Structure

This paper contains 64 sections, 96 theorems, 445 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Background on singular $SU(2)$ gauge theory
  3. Singular $SU(2)$ connections
  4. The Chern-Simons functional and flat connections
  5. The flip symmetry
  6. Perturbing the critical set
  7. Gradient trajectories and gradings
  8. Moduli spaces for cobordisms
  9. Reducible connections and negative definite pairs
  10. Counting critical points
  11. Orienting moduli spaces
  12. Instanton Floer homology groups for knots
  13. An analogue of Floer's instanton homology for knots
  14. The operators $\delta_1$ and $\delta_2$
  15. Holonomy operators and $v$-maps
  16. ...and 49 more sections

Key Result

Theorem 1

Let $K\subset Y$ be a knot embedded in an integer homology 3-sphere $Y$. The ${\bf Z}/4$-graded abelian group $I_\ast(Y,K)$ is an invariant of the equivalence class of the knot $(Y,K)$. Its Euler characteristic satisfies where $\lambda(Y)$ is the Casson invariant of $Y$ and $\sigma(K)$ is the signature of the knot $K\subset Y$.

Figures (17)

  • Figure 1: The relations (i) (left) and (ii) (right) of Proposition \ref{['prop:cobdelta']} .
  • Figure 2: On the left is depicted relation \ref{['eq:closedloop']}; on the right, the relation of Proposition \ref{['prop:vmap']}.
  • Figure 3: The relation of Proposition \ref{['prop:vmaprels2']}.
  • Figure 4:
  • Figure 5:
  • ...and 12 more figures

Theorems & Definitions (201)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Remark 2
  • Theorem 8
  • ...and 191 more