On the Categorified Wrapping Number Conjecture

TL;DR

This paper proves the Categorified Wrapping Number Conjecture for broad classes of annular links, providing a characterization of when resolutions yield nonzero homology classes based on crossing resolutions near trivial circles.

Contribution

It establishes the conjecture for large classes of annular links and characterizes resolution conditions for nonzero homology in the categorified setting.

Findings

Proved the conjecture for alternating annular links and tangle closures.

Characterized when resolutions produce nonzero homology classes.

Identified the role of crossing resolutions near trivial circles.

Abstract

We prove the Categorified Wrapping Number Conjecture for large classes of annular links, including alternating annular links and tangle closures exhibiting plumbed link phenomena. We do so by characterizing when a resolution is sufficient to produce a nonzero homology class in $k$-grading $\mathrm{wrap}(L)$ on its own. This characterization primarily concerns the type of crossing resolutions abutting trivial circles.