Instantons, special cycles, and knot concordance

TL;DR

This paper develops a new framework using equivariant singular instanton Floer theory to define and compute knot concordance invariants, leading to new topological insights and answers to existing conjectures.

Contribution

It introduces a unified framework for instanton-based concordance invariants, recovers recent invariants, and applies these to solve problems in knot and 3-manifold topology.

Findings

Computed the $s^lat$-invariant and fractional ideal invariants for two-bridge knots.

Proved a quasi-additivity property of the $s^lat$-invariant.

Produced infinite rank subgroups in the concordance group via satellite operations.

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 $s^\sharp$-invariant and fractional ideal invariants for two-bridge knots, and more. In particular, we prove a quasi-additivity property of $s^\sharp$, answering a question of Gong. We also introduce invariants that are formally similar to the Heegaard Floer $\tau$-invariant of Oszv\'ath and Szab\'o and the $\varepsilon$-invariant of Hom. We provide evidence for a precise relationship between these latter two invariants and the $s^\sharp$-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 Pinz\'on-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 quasi-positive 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.