The flip map and involutions on Khovanov homology

2025/06/01 by Da‐Ren Chen, Chen, Daren, Hongjian Yang +1 · 2 citations
Computer Science · Mathematics · #57K18 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2506.00824

openalex publication_date 2025/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The flip symmetry on knot diagrams induces an involution on Khovanov homology. We prove that this involution is determined by its behavior on unlinks; in particular, it is the identity map when working over \mathbbF₂. This confirms a folklore conjecture on the triviality of the Viro flip map. As a corollary, we prove that the symmetries on the transvergent and intravergent diagrams of a strongly invertible knot induce the same involution on Khovanov homology. We also apply similar techniques to study the half sweep-around map.

Citations

• Khovanov homology and equivariant surfaces
• Braiding on type A Soergel bimodules: semistrictness and naturality
• Involutive Khovanov homology and equivariant knots
• Involutions and the Chern-Simons filtration in instanton Floer homology
• Khovanov homology and the Involutive Heegaard Floer homology of branched double covers
• Khovanov homology of strongly invertible knots and their quotients
• A mixed invariant of non-orientable surfaces in equivariant Khovanov homology
• Fixing the functoriality of Khovanov homology: a simple approach
• Connected Floer homology of covering involutions
• Khovanov homology for links in #^r(S²× S¹)
• Involutive bordered Floer homology
• Field theories, stable homotopy theory and Khovanov homology
• A categorification of the stable SU(2) Witten-Reshetikhin-Turaev invariant of links in S2 x S1
• Fukaya categories of symmetric products and bordered Heegaard-Floer homology
• Matrix factorizations and link homology
• Holomorphic disks and topological invariants for closed three-manifolds
• Holomorphic disks and topological invariants for closed three-manifolds

Cited by

• Intrinsic Khovanov homology in \mathbbRP³
• A note on real Heegaard Floer homology and localization

Related

• Khovanov homology and the symmetry group of a knot
• Involutive Khovanov homology and equivariant knots
• Khovanov homology of strongly invertible knots and their quotients
• The Khovanov homology of alternating virtual links
• Knot invariants arising from homological operations on Khovanov homology