Transverse invariant as Khovanov skein spectrum at its extreme Alexander grading

2025/07/17 by Bhattacharyya, Nilangshu, Adithyan Pandikkadan, Pandikkadan, Adithyan · 1 citation
Medicine · Physics and Astronomy · #Algebraic Topology (math.AT) #Erythrocyte Function and Pathophysiology #FOS: Mathematics #Geometric Topology (math.GT) #Mechanical and Optical Resonators #Ophthalmology and Eye Disorders

paper · pdf · doi:10.48550/arxiv.2507.12882

openalex publication_date 2025/07/17 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28

Abstract

We develop a space-level formulation of Khovanov skein homology by constructing a stable homotopy type for annular links. We explicitly define a cover functor from the skein flow category to the cube flow category, thereby establishing the Khovanov skein spectrum. This spectrum extends the framework of Lipshitz and Sarkar's Khovanov spectrum and provides new avenues for understanding transverse link invariants in the annular setting. Furthermore, we establish a map from the Khovanov spectrum to the Khovanov skein spectrum, which, at extreme gradings, recovers the cohomotopy transverse invariant defined by Lipshitz, Ng, and Sarkar.

Citations

• Extreme Khovanov spectra
• A geometric description of the extreme Khovanov cohomology
• Heegaard Floer invariants of Legendrian knots in contact three--manifolds
• Transverse knots and Khovanov homology

Cited by

• Lipshitz--Sarkar stable homotopy type for certain planar trivalent graphs with perfect matchings

Related

• A generalized skein relation for Khovanov homology and a categorification of the θ-invariant
• Computing annular Khovanov homology
• Khovanov homology of strongly invertible knots and their quotients
• Transverse knots and Khovanov homology
• Crossing change on Khovanov homology and a categorified Vassiliev skein relation