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A condition on the Khovanov homology of three families of positive links

Lizzie Buchanan

TL;DR

This work addresses positivity obstructions for positive links by lifting diagram-independent bounds from the Jones polynomial to Khovanov homology. By analyzing extreme quantum gradings $j_{ ext{min}}(L)$ and $j_{ ext{max}}(L)$ and exploiting the invariant $p_1(L)$ (the absolute second Jones coefficient) together with the Conway polynomial, the authors derive explicit upper bounds on the maximal non-vanishing quantum degree $\bar{j}(L)$ of $Kh^{*,*}(L)$ for $p_1(L)\in\{0,1,2\}$. The main result shows that $\bar{j}(L)$ is bounded in terms of $\underline{j}(L)$, the number of components $n$, and the lead coefficient of the Conway polynomial, providing a Kh-based positivity obstruction that can be stronger than the Jones-based one in certain cases (with examples such as a non-positive quasipositive knot detected by Kh but not by Jones). The paper also discusses the limits of extending these bounds to broader classes like strongly quasipositive links and proposes open questions about extending Kh-based obstructions and identifying infinite families that differentiate positivity notions.

Abstract

In previous work, we developed diagram-independent upper bounds on the maximum degree of the Jones polynomial of three families of positive links. These families are characterized by the second coefficient of the Jones polynomial. In this paper, we extend those results and construct diagram-independent upper bounds on the maximum non-vanishing quantum degree of the Khovanov homology of three families of positive links. This can be used as a positivity obstruction.

A condition on the Khovanov homology of three families of positive links

TL;DR

This work addresses positivity obstructions for positive links by lifting diagram-independent bounds from the Jones polynomial to Khovanov homology. By analyzing extreme quantum gradings and and exploiting the invariant (the absolute second Jones coefficient) together with the Conway polynomial, the authors derive explicit upper bounds on the maximal non-vanishing quantum degree of for . The main result shows that is bounded in terms of , the number of components , and the lead coefficient of the Conway polynomial, providing a Kh-based positivity obstruction that can be stronger than the Jones-based one in certain cases (with examples such as a non-positive quasipositive knot detected by Kh but not by Jones). The paper also discusses the limits of extending these bounds to broader classes like strongly quasipositive links and proposes open questions about extending Kh-based obstructions and identifying infinite families that differentiate positivity notions.

Abstract

In previous work, we developed diagram-independent upper bounds on the maximum degree of the Jones polynomial of three families of positive links. These families are characterized by the second coefficient of the Jones polynomial. In this paper, we extend those results and construct diagram-independent upper bounds on the maximum non-vanishing quantum degree of the Khovanov homology of three families of positive links. This can be used as a positivity obstruction.

Paper Structure

This paper contains 5 sections, 8 theorems, 23 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Khovanov homology and Jones polynomial
  3. Khovanov homology and Jones polynomial
  4. Extreme Quantum Grading
  5. Further Exploration

Key Result

Theorem 1.1

Let $L$ be a positive link with Jones polynomial $V_L$. Let $p_1(L)$ be the absolute value of the second coefficient of $V_L$. If $p_1(L) = 0,$$1$, or $2$, then where $n$ is the number of link components and $\nabla_L$ is the Conway polynomial of $L$.

Figures (3)

  • Figure 1: Khovanov homology of $7_4$ (left), coefficient vector of unnormalized Jones polynomial $J_{7_4}(q)$ (middle), conversion to coefficient vector of Jones polynomial $V_{7_4}(t)$(right). Data from KnotAtlas KnotAtlas7-4.
  • Figure 2: Khovanov homology of $11_n57$ (left), conversion to coefficient vector of its Jones polynomial (right). Data from KnotAtlas KnotAtlas11n57.
  • Figure 3: A crossing (left), its $A$-smoothing (middle), and $B$-smoothing (right)

Theorems & Definitions (16)

  • Theorem 1.1: Buchanan_2022Buchanan2023buchanan2025conditionjonespolynomialfamily
  • Theorem 1.2
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Proposition 3.6: GMMS2018
  • Corollary 3.7
  • Lemma 3.8: Lickorish
  • ...and 6 more