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On the Categorified Wrapping Number Conjecture

Benjamin Daniels, Melissa Zhang

TL;DR

The paper tackles Grigsby's Categorified Wrapping Number Conjecture for annular links by introducing perfectly wrapped uniform resolutions and a local-to-global analysis that identifies when a single resolution yields a nonzero $k$-graded class in annular Khovanov homology. The authors develop a precise framework distinguishing types of trivial circles (type $0$ and type $1$), insulation, and the kernel/image behavior to certify nonvanishing at the wrapping number, enabling proofs for alternating annular links and several plumbed-tangle classes. A key contribution is converting almost uniform resolutions into perfectly wrapped uniform ones, broadening the scope to many link families and preserving the pwur property under cabling and tangle operations. The results yield a lower bound on the rank of $ ext{AKh}(L; ext{wrap}(L))$ via the count $U(L)$ of pwur resolutions and significantly advance the categorified skein-theory understanding of annular links, extending Hoste–Przytycki-type phenomena to the categorified setting.

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.

On the Categorified Wrapping Number Conjecture

TL;DR

The paper tackles Grigsby's Categorified Wrapping Number Conjecture for annular links by introducing perfectly wrapped uniform resolutions and a local-to-global analysis that identifies when a single resolution yields a nonzero -graded class in annular Khovanov homology. The authors develop a precise framework distinguishing types of trivial circles (type and type ), insulation, and the kernel/image behavior to certify nonvanishing at the wrapping number, enabling proofs for alternating annular links and several plumbed-tangle classes. A key contribution is converting almost uniform resolutions into perfectly wrapped uniform ones, broadening the scope to many link families and preserving the pwur property under cabling and tangle operations. The results yield a lower bound on the rank of via the count of pwur resolutions and significantly advance the categorified skein-theory understanding of annular links, extending Hoste–Przytycki-type phenomena to the categorified setting.

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 -grading on its own. This characterization primarily concerns the type of crossing resolutions abutting trivial circles.
Paper Structure (14 sections, 24 theorems, 29 equations, 20 figures)

This paper contains 14 sections, 24 theorems, 29 equations, 20 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Annular Khovanov homology
  4. Cube of resolutions
  5. The annular Khovanov chain complex
  6. Wrapping number of diagrams and resolutions
  7. Perfectly wrapped and uniform resolutions
  8. Terminology and notation
  9. Type $0$ and type $1$ trivial circles
  10. Proof of the main result
  11. An expanded class of diagrams admitting perfectly wrapped uniform resolutions
  12. Classes of links admitting perfectly wrapped uniform resolutions
  13. Alternating links
  14. Other classes and modifications of links

Key Result

Theorem 3

Let $L$ be an annular link with a diagram $D$ admitting a perfectly wrapped uniform resolution $D_u$. Then $\mathrm{AKh}(L)$ is nontrivial in $k$-grading $\mathrm{wrap}(L)$.

Figures (20)

  • Figure 1: Annular closures of these tangles are examples of the two classes of annular links from Corollary \ref{['cor:buildingclasseswithpwur']}.
  • Figure 2: Examples of operations from Corollary \ref{['cor:linkopsobtainingmorepwur']}.
  • Figure 3: Left column, from top to bottom: merges $\mathcal{W} \sqcup \mathcal{W} \rightarrow \mathcal{W}$, $\mathcal{V} \sqcup \mathcal{W} \rightarrow \mathcal{V}$, and $\mathcal{V} \sqcup \mathcal{V} \rightarrow \mathcal{W}$, respectively. Right column, from top to bottom: splits $\mathcal{W} \rightarrow \mathcal{W} \sqcup \mathcal{W}$, $\mathcal{V} \rightarrow \mathcal{V} \sqcup \mathcal{W}$, and $\mathcal{W} \rightarrow \mathcal{V} \sqcup \mathcal{V}$.
  • Figure 4: A pair of pants cobordism, which represents a merging of two circles when read from top to bottom. The red arc is the surgery arc on the 0-resolution; the blue arc is the dual surgery arc on the 1-resolution.
  • Figure 5: A dots-and-arrows depiction of a $\mathbb{Z}$-module merge map $m: \mathbb{V} \otimes \mathbb{V} \to \mathbb{W}$. There are two arrows $a,b$ with sources $v_+ \otimes v_-$ and $v_- \otimes v_+$, respectively. The target of both arrows is $w_-$.
  • ...and 15 more figures

Theorems & Definitions (61)

  • Conjecture 1
  • Conjecture 2
  • Theorem 3
  • Theorem 4
  • Corollary 5
  • Corollary 6
  • Corollary 7
  • Remark 10
  • Remark 11
  • Remark 12
  • ...and 51 more