Insensitivity of Khovanov homology under rim surgery

2026/07/28 by Qiuyu Ren, Fang-Rong Zhan
Mathematics · #math.GT #math.QA

paper · pdf

Abstract

We show that rim surgery on a smooth, oriented, properly embedded surface in B⁴ does not change the map on Khovanov homology induced by the surface, answering a question raised by Hayden and Sundber. More generally, the same insensitivity holds for a class of local surgery operations that we call local annulus replacements. Combined with the work of Hayden--Sundberg, this result yields exotic pairs of surfaces in B⁴ that cannot be related by any sequence of local annulus replacements. We give two proofs, both making essential use of Khovanov skein lasagna modules.

Citations

• Khovanov skein lasagna modules with 1-dimensional inputs
• Khovanov homology and exotic 4-manifolds
• Skein lasagna modules and handle decompositions
• Khovanov homology and exotic surfaces in the 4-ball
• A categorification of the stable SU(2) Witten-Reshetikhin-Turaev invariant of links in S2 x S1
• Knotted surfaces in 4-manifolds
• Surfaces in 4-Manifolds

Related

• An Extension of Khovanov Homology to Immersed Surface Cobordisms
• Khovanov homology and exotic surfaces in the 4-ball
• Surgery obstructions from Khovanov homology
• Surfaces in 4-Manifolds
• A mixed invariant of non-orientable surfaces in equivariant Khovanov homology