Annular Khovanov homology and meridional disks

TL;DR

This paper compares annular Khovanov homology and Floer-theoretic invariants for infinite families of annular links, revealing divergent growth behaviors and providing evidence for the wrapping conjecture.

Contribution

It demonstrates the existence of infinite families where Khovanov gradings grow unbounded while Floer gradings remain bounded, and extends spectral sequences to annular settings.

Findings

Khovanov gradings grow infinitely large for certain links.

Floer-theoretic gradings are bounded in the same cases.

Supports the wrapping conjecture and its categorified version.

Abstract

We exhibit infinite families of annular links for which the maximum non-zero annular Khovanov grading grows infinitely large but the maximum non-zero annular Floer-theoretic gradings are bounded. We also show this phenomenon exists at the decategorified level for some of the infinite families. Our computations provide further evidence for the wrapping conjecture of Hoste-Przytycki and its categorified analogue. Additionally, we show that certain satellite operations cannot be used to construct counterexamples to the categorified wrapping conjecture. We also extend the Batson-Seed link splitting spectral sequence to the setting of annular Khovanov homology.