Involutive Khovanov homology and equivariant knots

Abstract

For strongly invertible knots, we define an involutive version of Khovanov homology, and from it derive a pair of integer-valued invariants $(\underline{s},\overline{s})$, which is an equivariant version of Rasmussen's $s$-invariant. Using these invariants, we reprove that the infinite family of knots $J_n$ introduced by Hayden each admits exotic pairs of slice disks. Our construction is intended to give a Khovanov-theoretic analogue of the formalism given by Dai, Mallick and Stoffregen in involutive knot Floer theory.

arXiv comment57 pages. Revision in v3: fixed the definition of the generalized crossing change map. v4: Added computational results. v5: Fixed error in Section 3.1 (see attached Erratum)

References in corpus (9)

• Khovanov's homology for tangles and cobordisms
• Involutive Heegaard Floer homology
• Connected sums and involutive knot Floer homology
• The Bar-Natan homology and unknotting number
• Positive Links
• A refinement of Khovanov homology
• Divisibility of Lee's class and its relation with Rasmussen's invariant
• A cobordism realizing crossing change on ${\mathfrak{sl}}_2$ tangle homology and a categorified Vassiliev skein relation
• A family of slice-torus invariants from the divisibility of Lee classes