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Searching for non-order-preserving braids algorithmically

Jonathan Johnson, Nancy Scherich, Hannah Turner

Abstract

An $n$-strand braid is order-preserving if its action on the free group $F_n$ preserves some bi-order of $F_n$. A braid $β$ is order-preserving if and only if the link $L$ obtained as the union of the closure of $β$ and its axis has bi-orderable complement. We describe and implement an algorithm which, given a non-order-preserving braid $β$, confirms this property and returns a proof that $β$ is indeed not order-preserving. Guided by the algorithm, we prove that the infinite family of simple 3-braids $σ_1σ_2^{2m+1}$ are not order-preserving for any integer $m$.

Searching for non-order-preserving braids algorithmically

Abstract

An -strand braid is order-preserving if its action on the free group preserves some bi-order of . A braid is order-preserving if and only if the link obtained as the union of the closure of and its axis has bi-orderable complement. We describe and implement an algorithm which, given a non-order-preserving braid , confirms this property and returns a proof that is indeed not order-preserving. Guided by the algorithm, we prove that the infinite family of simple 3-braids are not order-preserving for any integer .

Paper Structure

This paper contains 9 sections, 19 theorems, 22 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Organization of the paper.
  3. Acknowledgments
  4. Braids and orders
  5. Precones
  6. Words with zero exponent sum
  7. Algorithms
  8. Implementation
  9. A Family of Non-Order-Preserving Braids

Key Result

Proposition 1

An $n$-strand braid $\beta$ is order-preserving if and only if $\beta$ preserves a $k$-precone of the free group $F_n$ for every positive integer $k$.

Figures (3)

  • Figure 1: (A) The closure $\widehat{\beta}$ of of a 3-braid $\beta$ together with its axis $a$ forms a braided link. (B) The Artin generator $\sigma_i$.
  • Figure 2: Our conventions for the induced action of the $n$-strand braid $\sigma_1$ on the generators of $\pi_1(D_n)$.
  • Figure 3: This tree depicts how Algorithm 30 obstructs order-preservingness of $\sigma_1\sigma_2^{-3}$. For conjugation, we use the notation $g^h:=hgh^{-1}$ for two elements $g,h\in F_n$.

Theorems & Definitions (38)

  • Proposition 1
  • Theorem 2
  • Theorem 3
  • Theorem 5
  • Definition 7
  • Definition 8
  • Remark 9
  • Proposition 10: KR-BraidsOrderings
  • Definition 11
  • Definition 12
  • ...and 28 more