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A slice Cromwell inequality of homogeneous links

Tetsuya Ito

TL;DR

The paper generalizes Cromwell's inequality for homogeneous links from the 3D Euler characteristic $\chi(L)$ to the 4D Euler characteristic $\chi_4(L)$, proving a slice Cromwell inequality $\min \deg_v P_L(v,z) \le 1-\chi_4(L)$ and a sharpened diagram-level bound expressed via Seifert-graph data $s(D)$, $w(D)$, $s_+(D)$, and $\#_{sp}L$. It accomplishes this by a skein-tree analysis of the HOMFLY polynomial and a detailed accounting of Seifert-graph blocks, leading to a noncanceling leading term tied to $z^{1-\chi(L)}$ and monomial $v^{ -s(D)+w(D)+2s_+(D)+1-2\#_{sp}L}$. The results connect to slice-torus invariants $\phi$, yielding upper bounds that refine Bennequin-type inequalities and provide a framework for equality conjectures, particularly regarding positivity and quasipositive homogeneous links. Additionally, for alternating links, the authors derive a bound relating the minimum $v$-degree to the signature $\sigma(L)$, reinforcing the connection between classical knot invariants and HOMFLY-analytic data.

Abstract

Cromwell proved that the minimum $v$-degree of the HOMFLY polynomial of homogeneous link $L$ is bounded above by $1-χ(L)$, where $χ(L)$ is the maximum Euler characteristic of Seifert surfaces of $L$. We prove its slice version, stating that the minimum $v$-degree of the HOMFLY polynomial of homogeneous link $L$ is bounded above by $1-χ_4(L)$, the maximum 4-dimensional Euler characteristic of $L$. As a byproduct, we prove a conjecture of Stoimenow that for an alternating link, the minimum $v$-degree of the HOMFLY polynomial is smaller than or equal to its signature.

A slice Cromwell inequality of homogeneous links

TL;DR

The paper generalizes Cromwell's inequality for homogeneous links from the 3D Euler characteristic to the 4D Euler characteristic , proving a slice Cromwell inequality and a sharpened diagram-level bound expressed via Seifert-graph data , , , and . It accomplishes this by a skein-tree analysis of the HOMFLY polynomial and a detailed accounting of Seifert-graph blocks, leading to a noncanceling leading term tied to and monomial . The results connect to slice-torus invariants , yielding upper bounds that refine Bennequin-type inequalities and provide a framework for equality conjectures, particularly regarding positivity and quasipositive homogeneous links. Additionally, for alternating links, the authors derive a bound relating the minimum -degree to the signature , reinforcing the connection between classical knot invariants and HOMFLY-analytic data.

Abstract

Cromwell proved that the minimum -degree of the HOMFLY polynomial of homogeneous link is bounded above by , where is the maximum Euler characteristic of Seifert surfaces of . We prove its slice version, stating that the minimum -degree of the HOMFLY polynomial of homogeneous link is bounded above by , the maximum 4-dimensional Euler characteristic of . As a byproduct, we prove a conjecture of Stoimenow that for an alternating link, the minimum -degree of the HOMFLY polynomial is smaller than or equal to its signature.

Paper Structure

This paper contains 2 sections, 9 theorems, 21 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem

Key Result

Theorem 1

For a homogeneous linkIn cr link is always assumed to be non-split and the theorem is proved for non-split links, but one can check that the theorem applies for non-split links.$L$, holds. Furthermore, the equality holds if and only if $L$ is positive.

Figures (2)

  • Figure 1: (i) Homegenous diagram (ii) Seifert graph (iii) The blocks of Seifert graph
  • Figure 2: Skein resolution tree

Theorems & Definitions (17)

  • Theorem 1: Cromwell inequality
  • Theorem 2
  • Theorem 3
  • Corollary 1: Slice Cromwell inequality
  • Conjecture 1
  • Theorem 4
  • proof
  • Theorem 5
  • Definition 6: Homogeneous graphs and links
  • Theorem 7: Cromwell cr
  • ...and 7 more