Connected sums and involutive knot Floer homology

Abstract

We prove a formula for the conjugation action on the knot Floer complex of the connected sum of two knots. Using the formula we construct a homomorphism from the smooth concordance group to an abelian group consisting of chain complexes with homotopy automorphisms, modulo an equivalence relation. Using our connected sum formula, we perform some example computations of Hendricks and Manolescu's involutive invariants on large surgeries of connected sums of knots.

Minor expository improvements. To appear, Proceedings of the London Mathematical Society

References in corpus (10)

• Floer homology and knot complements
• Lens space surgeries and L-space homology spheres
• Contact structures, sutured Floer homology and TQFT
• Involutive Heegaard Floer homology
• Link cobordisms and functoriality in link Floer homology
• Quasi-stabilization and basepoint moving maps in link Floer homology
• Graph cobordisms and Heegaard Floer homology
• Pin(2)-equivariant Seiberg-Witten Floer homology of Seifert fibrations
• $\mathrm{Pin}(2)$-monopole Floer homology, higher compositions and connected sums
• Involutive Heegaard Floer homology and rational cuspidal curves

Cited by in corpus (12)

• Link cobordisms and functoriality in link Floer homology
• More concordance homomorphisms from knot Floer homology
• Duality and mapping tori in Heegaard Floer homology
• Linear independence of rationally slice knots
• Involutive Heegaard Floer homology and rational cuspidal curves
• On sums of torus knots concordant to alternating knots
• A mnemonic for the Lipshitz-Ozsváth-Thurston correspondence
• Locally equivalent Floer complexes and unoriented link cobordisms
• Upsilon type concordance invariants
• On the nonorientable four-ball genus of torus knots
• Heegaard Floer homology and plane curves with non-cuspidal singularities
• Involutive Khovanov homology and equivariant knots