Framed instanton homology and concordance

Abstract

We define two concordance invariants of knots using framed instanton homology. These invariants ${\nu }^{♯}$ and ${\tau }^{♯}$ provide bounds on slice genus and maximum self-linking number, and the latter is a concordance homomorphism which agrees in all known cases with the $\tau$ invariant in Heegaard Floer homology. We use ${\nu }^{♯}$ and ${\tau }^{♯}$ to compute the framed instanton homology of all nonzero rational Dehn surgeries on: 20 of the 35 nontrivial prime knots through 8 crossings, infinite families of twist and pretzel knots, and instanton L-space knots; and of 19 of the first 20 closed hyperbolic manifolds in the Hodgson--Weeks census. In another application, we determine when the cable of a knot is an instanton L-space knot. Finally, we discuss applications to the spectral sequence from odd Khovanov homology to the framed instanton homology of branched double covers, and to the behaviors of ${\tau }^{♯}$ and $\tau$ under genus-2 mutation.

64 pages, 15 figures; v2: added another family of examples; v3: accepted version, to appear in Journal of Topology

References in corpus (2)

• On the Khovanov and knot Floer homologies of quasi-alternating links
• Immersed curves in Khovanov homology

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• Instanton Floer homology, sutures, and Euler characteristics
• Small Dehn surgery and SU(2)
• Knot surgery formulae for instanton Floer homology II: applications
• Instantons, special cycles, and knot concordance
• SU(2) representations and a large surgery formula
• A family of slice-torus invariants from the divisibility of Lee classes
• Involutions, knots, and Floer K-theory
• Framed instanton homology and concordance, II
• Knot surgery formulae for instanton Floer homology I: the main theorem