Instantons, special cycles, and knot concordance

TL;DR

The paper develops a unified framework linking equivariant singular instanton theory with a Chern–Simons filtration to define and compare knot concordance invariants. It introduces $\ ext{S}$-complexes and their equivariant/framed variants, then constructs special cycles to extract numerical invariants such as $\widetilde{s}$, $s^{\sharp}$, and $\widetilde{\varepsilon}$, proving their relations to classical invariants and their quasi-additivity properties. For two-bridge knots, the framework yields explicit computations and shows the invariants are determined by the knot signature, connecting to existing KM invariants and Floer-theoretic conjectures. The work extends to general holonomy parameters via enriched and filtered complexes, enabling applications to satellites, the homology cobordism group, and the existence of non-abelian $SU(2)$ representations on concordance complements, thereby broadening the toolkit for 4-manifold knot concordance problems.

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 $τ$-invariant of Oszváth and Szabó 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 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 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.

Figures (11)

• Figure 1: Tangle corresponding to a pattern $P$ determined by $\{a_i\}_{i=1}^m$ and $\{b_i\}_{i=1}^n$.
• Figure 2: The ${{\mathbb Q}[\![\Lambda]\!]}$-module $H(C^\sharp)$ in the proof of \ref{['prop:twobridgessharp']}.
• Figure 3: Two-bridge knot $K_{m, n}=K(212mn-68n+53, 106m-34)$. In the box labelled "$-m$" there are $m$ full negative twists, and in the box labelled "$n$" there are $n$ full positive twists.
• Figure 4: A Seifert surface for $K_{m, n}$ and its symplectic basis.
• Figure 5: Changing $n+3$ negative crossings in the diagram for $K_{m,n}$ produces an unknot.

Theorems & Definitions (99)

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