Involutive Khovanov homology and equivariant knots

TL;DR

This work develops an involutive Khovanov theory for strongly invertible knots and defines equivariant Rasmussen invariants $\underline{s}$ and $\overline{s}$ that bound the ordinary $s$-invariant. By constructing the involutive complex and its reduced variants, and proving robust invariance and cobordism properties, the authors connect Khovanov theory to equivariant concordance via Lee-class divisibility, mirroring involutive knot Floer theory. The main result applies these invariants to the Hayden family $J_n$, proving the existence of exotic pairs of slice disks, and demonstrates the utility of the equivariant framework for distinguishing equivariant slice phenomena. The framework further extends to $2$-periodic links, broadening the reach of involutive and equivariant Khovanov-type invariants and providing new computational data for small knots.

Abstract

For strongly invertible knots, we define an involutive version of Khovanov homology, and from it derive a pair of integer-valued invariants $(\underline{s}, \bar{s})$, 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.

Figures (12)

• Figure 1: The knot $J_n$ and the two slice disks $D_n, D'_n$ obtained by compressing along the two colored circles.
• Figure 2: Involutive Reidemeister moves
• Figure 3: $F$ and $F^\tau$.
• Figure 4: Modifying the R1-move
• Figure 5: $ab$-coloring on the Seifert circles of $K$.

Theorems & Definitions (165)

• Theorem 1
• Definition 1.1
• Theorem 2
• Theorem 3
• Proposition 1.2
• Proposition 1.3
• Proposition 1.4
• Proposition 1.5
• Proposition 1.6
• Proposition 1.7