Transverse invariant as Khovanov skein spectrum at its extreme Alexander grading

TL;DR

This paper constructs a stable homotopy type for annular links' Khovanov skein homology, extending previous spectra and connecting to transverse invariants at extreme Alexander gradings.

Contribution

It introduces a space-level formulation of Khovanov skein homology for annular links and establishes a spectrum that generalizes Lipshitz and Sarkar's work, linking to transverse invariants.

Findings

Defined a cover functor from skein flow to cube flow category.

Constructed the Khovanov skein spectrum for annular links.

Connected the spectrum to transverse invariants at extreme gradings.

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.