Instantons, special cycles and knot concordance

Daemi, Aliakbar; Imori, Hayato; Sato, Kouki; Scaduto, Christopher; Taniguchi, Masaki

doi:10.2140/gt.2025.29.4189

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Instantons, special cycles and knot concordance

Aliakbar Daemi, Hayato Imori, Kouki Sato, Christopher Scaduto and Masaki Taniguchi

Geometry & Topology 29 (2025) 4189–4298

DOI: 10.2140/gt.2025.29.4189

Abstract

We introduce a framework for defining concordance invariants of knots using equivariant singular instanton Floer theory with Chern–Simons filtration. It is demonstrated that many of the concordance invariants defined using instantons in recent years can be recovered from our framework. This relationship allows us to compute Kronheimer and Mrowka’s $s^{♯}$ -invariant and fractional ideal invariants for two-bridge knots, and more. In particular, we prove a quasiadditivity property of $s^{♯}$ , answering a question of Gong. We also introduce invariants that are formally similar to the Heegaard Floer $τ$ -invariant of Ozsváth and Szabó and the $𝜀$ -invariant of Hom. We provide evidence for a precise relationship between these latter two invariants and the $s^{♯}$ -invariant.

Some new topological applications that follow from our techniques are as follows. First, we produce a wide class of patterns whose induced satellite maps on the concordance group have the property that their images have infinite rank, giving a partial answer to a conjecture of Hedden and Pinzón-Caicedo. Second, we produce infinitely many two-bridge knots $K$ which are torsion in the algebraic concordance group and yet have the property that the set of positive $1 ∕ n$ -surgeries on $K$ is a linearly independent set in the homology cobordism group. Finally, for a knot which is quasipositive and not slice, we prove that any concordance from the knot admits an irreducible $SU (2)$ -representation on the fundamental group of the concordance complement.

While much of the paper focuses on constructions using singular instanton theory with the traceless meridional holonomy condition, we also develop an analogous framework for concordance invariants in the case of arbitrary holonomy parameters, and some applications are given in this setting.

Keywords

concordance invariants, instantons, equivariant singular instanton homology, special cycles

Mathematical Subject Classification

Primary: 57K18, 57R58

References

Publication

Received: 20 June 2023

Revised: 11 July 2024

Accepted: 28 September 2024

Published: 26 November 2025

Proposed: Simon Donaldson

Seconded: Tomasz S Mrowka, András I Stipsicz

Authors

Aliakbar Daemi
Department of Mathematics and Statistics Washington University in St Louis St Louis, MO United States

Hayato Imori
Department of Mathematics Graduate School of Science Kyoto University Kyoto Japan	Department of Mathematics Korea Advanced Institute of Science and Technology Daejeon South Korea

Kouki Sato
Meijo University Nagoya Japan

Christopher Scaduto
Department of Mathematics University of Miami Coral Gables, FL United States

Masaki Taniguchi
Department of Mathematics Graduate School of Science Kyoto University Kyoto Japan

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Volume 29, issue 8 (2025)