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On the negative band number

Michele Capovilla-Searle, Tetsuya Ito, Keiko Kawamuro, Rebecca Sorsen

TL;DR

This work defines and investigates the negative band number ${\tt nb}$ for braids, knots, and links through the Birman-Ko-Lee left-canonical form, establishing a filtration on $B_n$ that categorizes strongly quasipositive and almost strongly quasipositive braids. It provides precise characterizations of SQP and ASQP braids in terms of the summit structures ${\tt SS}$ and ${\tt SSS}$, with polynomial-time criteria for small braid indices and conditional results for larger $n$. The paper derives bounds on ${\tt nb}(\beta)$ from ${\tt LCF}(\beta)$, analyzes the relationship between minimal words and ${\tt nb}$, and studies how ${\tt nb}$ behaves under stabilization, including a construction showing arbitrarily large stabilization required to realize ${\tt nb}(K)$. These findings connect braid-theoretic invariants to low-dimensional topology and contact geometry, clarifying the computational landscape of SQP/ASQP detection and linking ${\tt nb}$ to the Bennequin framework. Overall, the results advance both the theoretical understanding and algorithmic handling of negativity in braid representations and their geometric/topological consequences.

Abstract

We study the negative band number of braids, knots, and links using Birman, Ko, and Lee's left-canonical form of a braid. As applications, we characterize up to conjugacy strongly quasipositive braids and almost strongly quasipositive braids.

On the negative band number

TL;DR

This work defines and investigates the negative band number for braids, knots, and links through the Birman-Ko-Lee left-canonical form, establishing a filtration on that categorizes strongly quasipositive and almost strongly quasipositive braids. It provides precise characterizations of SQP and ASQP braids in terms of the summit structures and , with polynomial-time criteria for small braid indices and conditional results for larger . The paper derives bounds on from , analyzes the relationship between minimal words and , and studies how behaves under stabilization, including a construction showing arbitrarily large stabilization required to realize . These findings connect braid-theoretic invariants to low-dimensional topology and contact geometry, clarifying the computational landscape of SQP/ASQP detection and linking to the Bennequin framework. Overall, the results advance both the theoretical understanding and algorithmic handling of negativity in braid representations and their geometric/topological consequences.

Abstract

We study the negative band number of braids, knots, and links using Birman, Ko, and Lee's left-canonical form of a braid. As applications, we characterize up to conjugacy strongly quasipositive braids and almost strongly quasipositive braids.

Paper Structure

This paper contains 11 sections, 26 theorems, 56 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Band generators
  4. Canonical factors
  5. The left-canonical form
  6. Summit set and super summit set
  7. Reduction and the shortest word problem
  8. Detection of SQP and ASQP braids
  9. Bounds of ${\tt nb}(\beta)$ via ${\tt LCF}(\beta)$
  10. Shortest words and ${\tt nb}(\beta)$
  11. Behavior of ${\tt nb}(\beta)$ under stabilization

Key Result

Lemma 1.2

Let $n\geq 3$. A braid $\beta\in B_n$ is conjugate to a SQP braid, if and only if every element $\beta' \in {\tt SSS}([\beta])$ has $\inf(\beta')\geq 0$, if and only if there exists an element $\beta' \in {\tt SSS}([\beta])$ with $\inf(\beta')\geq 0$. Thus, given an $n$-braid one can determine wheth

Figures (10)

  • Figure 1: Band generators for $B_4$. The braid strands are numbered from 1 (bottom strand) to 4 (top strand).
  • Figure 2: The partially ordered set $({\tt CnFct}(B_4), \prec)$. The line connecting a lower diagram $A$ and an upper diagram $B$ means $A \prec B$.
  • Figure 3: The effect of a negative tunnel-stabilization.
  • Figure 4: From (1) to (3) the pink strand is pinched and dragged through the tunnel. From (3) to (4) the strand is flipped performing a Reidemeister I move introducing the stabilization. From (4) to (5) the tunnel is being shortened performing a Reidemeister move II between the top and bottom strand. From (5) to (6) the tunnel is represented by two bands positive and negative bands.
  • Figure 5: (Top) The word $\gamma_j$. (Middle) The braid $\beta_4$. (Bottom) Part of the Seifert surface $\Sigma_n$ of $L_n$ involving $\gamma_j$.
  • ...and 5 more figures

Theorems & Definitions (58)

  • Definition 1.1
  • Lemma 1.2: Lemma \ref{['cor:sqpconj']}, Corollary \ref{['cor:SQP problem']}
  • Theorem 1.3: Theorem \ref{['theorem:summit']}, Corollary \ref{['cor:ASQP problem']}
  • Theorem 1.4: Theorem \ref{['thm:stronger-version']}
  • Conjecture 1.5
  • Theorem 1.6: Theorem \ref{['thm:3braid-nb']}
  • Theorem 1.7: Theorem \ref{['lem:nb(beta)']}
  • Theorem 1.8: Theorem \ref{['prop:stabilization example']}
  • Definition 2.1
  • Definition 2.2
  • ...and 48 more