Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The slice-Bennequin inequality for the fractional Dehn twist coefficient

Peter Feller

TL;DR

The paper characterizes the Fractional Dehn Twist Coefficient ${\omega}$ on the braid group as the unique homogeneous quasimorphism of defect at most $1$ with ${\omega}(\Delta^2)=1$ and ${\omega}(B_{n-1})=0$, and it proves a slice-Bennequin type inequality for ${\omega}$, giving an affine linear lower bound for the smooth slice genus in terms of the braid's ${\omega}$. The approach ties ${\omega}$ to the homogenization of Upsilon and leverages cobordism arguments and Dehornoy order to relate braid invariants to 4-dimensional knot invariants. The authors also show that many other homogeneous quasimorphisms with similar normalization exist via Bestvina–Fujiwara-type constructions, while a unifying appendix places the slice-Bennequin inequality and its ${\omega}$-counterpart in a common homogenization framework. Overall, the work strengthens connections between braid group quasimorphisms, concordance, and 4-manifold topology, and provides a robust toolkit for bounding slice genus from braid data.

Abstract

We characterize the fractional Dehn twist coefficient (FDTC) on the $n$-stranded braid group as the unique homogeneous quasimorphism to the real numbers of defect at most 1 that equals 1 on the positive full twist and vanishes on the $(n-1)$-stranded braid subgroup. In a different direction, we establish that the slice-Bennequin inequality holds with the FDTC in place of the writhe. In other words, we establish an affine linear lower bound for the smooth slice genus of the closure of a braid in terms of the braid's FDTC. We also discuss connections between these two seemingly unrelated results. In the appendix we provide a unifying framework for the slice-Bennequin inequality and its counterpart for the FDTC.

The slice-Bennequin inequality for the fractional Dehn twist coefficient

TL;DR

The paper characterizes the Fractional Dehn Twist Coefficient on the braid group as the unique homogeneous quasimorphism of defect at most with and , and it proves a slice-Bennequin type inequality for , giving an affine linear lower bound for the smooth slice genus in terms of the braid's . The approach ties to the homogenization of Upsilon and leverages cobordism arguments and Dehornoy order to relate braid invariants to 4-dimensional knot invariants. The authors also show that many other homogeneous quasimorphisms with similar normalization exist via Bestvina–Fujiwara-type constructions, while a unifying appendix places the slice-Bennequin inequality and its -counterpart in a common homogenization framework. Overall, the work strengthens connections between braid group quasimorphisms, concordance, and 4-manifold topology, and provides a robust toolkit for bounding slice genus from braid data.

Abstract

We characterize the fractional Dehn twist coefficient (FDTC) on the -stranded braid group as the unique homogeneous quasimorphism to the real numbers of defect at most 1 that equals 1 on the positive full twist and vanishes on the -stranded braid subgroup. In a different direction, we establish that the slice-Bennequin inequality holds with the FDTC in place of the writhe. In other words, we establish an affine linear lower bound for the smooth slice genus of the closure of a braid in terms of the braid's FDTC. We also discuss connections between these two seemingly unrelated results. In the appendix we provide a unifying framework for the slice-Bennequin inequality and its counterpart for the FDTC.
Paper Structure (6 sections, 9 theorems, 38 equations, 1 figure)

This paper contains 6 sections, 9 theorems, 38 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Context for Theorem \ref{['thm:sBineq']}: Bennequin and slice-Bennequin inequalities
  3. The proof of Theorem \ref{['thm:CharPropForFDTC']}
  4. Constructions of quasimorphisms and the proof of Proposition \ref{['prop:uncqms']}
  5. The proofs of Theorem \ref{['thm:sBineq']} and Proposition \ref{['prop:Df<=A(n-1)']}
  6. The slice-Bennequin inequality for the homogenization of concordance homomorphisms

Key Result

Theorem 1

For every integer $n\geq 3$, there exists a unique homogeneous quasimorphism ${\omega}\colon B_n\to \mathbb{R}$ with defect at most 1 that satisfies the following properties:

Figures (1)

  • Figure 1: Isotopies and cobordisms proving Lemma \ref{['lem:cobordisms']}. For readability of the diagrams the illustration is for $n=4$.

Theorems & Definitions (21)

  • Theorem 1
  • Proposition 2
  • Theorem 3
  • Proposition 5
  • Corollary 6
  • proof : Proof of Corollary \ref{['cor:defectvssB']}
  • Remark 9
  • proof : Proof of Theorem \ref{['thm:CharPropForFDTC']}
  • Lemma 10
  • proof
  • ...and 11 more