Instantons, special cycles, and knot concordance

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 quasi-additivity property of $s^{♯}$, answering a question of Gong. We also introduce invariants that are formally similar to the Heegaard Floer $\tau$-invariant of Oszváth and Szabó and the $\epsilon$-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 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.

arXiv comment112 pages, 11 figures, 1 table

References in corpus (23)

• Knot Floer homology and the four-ball genus
• Knot Floer homology and rational surgeries
• Knot homology groups from instantons
• Computations of the Ozsvath-Szabo knot concordance invariant
• Gauge theory and Rasmussen's invariant
• Bordered Heegaard Floer homology and the tau-invariant of cable knots
• Seifert forms and concordance
• Rasmussen's spectral sequences and the sl(N)-concordance invariants
• Framed instanton homology and concordance
• A characterization of quasipositive Seifert surfaces (Constructions of quasipositive knots and links, III)
• Pin(2)-equivariant Seiberg-Witten Floer homology of Seifert fibrations
• Satellites of Infinite Rank in the Smooth Concordance Group
• Filtered instanton Floer homology and the homology cobordism group
• Chern-Simons functional, singular instantons, and the four-dimensional clasp number
• Chern-Simons Functional and the Homology Cobordism Group
• Topologically slice knots that are not smoothly slice in any definite 4-manifold
• The ${\nu }^{+}$-equivalence classes of genus one knots
• Instantons and some concordance invariants of knots
• Divisibility of Lee's class and its relation with Rasmussen's invariant
• Equivariant aspects of singular instanton Floer homology
• Knot concordance invariants from Seiberg-Witten theory and slice genus bounds in 4-manifolds
• On independence of iterated Whitehead doubles in the knot concordance group
• Instanton knot invariants with rational holonomy parameters and an application for torus knot groups

Cited by in corpus (1)

• Cables of the figure-eight knot via real Frøyshov invariants