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Studying links via plats: split and composite links

William W. Menasco, Deepisha Solanki

Abstract

Our main results concern changing an arbitrary plat presentation of a split or composite link to one which is obviously recognizable as being split or composite. Pocket moves, first described in \cite{unlinkviaplats}, are utilized -- a pocket move alters a plat presentation without changing its link type, its bridge index or the double coset. A plat presentation of a split link is split if the planar projection of the plat presentation is not connected. We prove that pocket moves are the only obstruction to representing split links by split plat presentations. Since any pocket move corresponds to a sequence of double coset moves, we have the corollary that the double coset of every plat presentation of a split link has a split plat presentation. We obtain an analogous result for composite links by utilizing flip moves, which were also first described in the second author's work, arXiv:2308.00732 [math.GT].

Studying links via plats: split and composite links

Abstract

Our main results concern changing an arbitrary plat presentation of a split or composite link to one which is obviously recognizable as being split or composite. Pocket moves, first described in \cite{unlinkviaplats}, are utilized -- a pocket move alters a plat presentation without changing its link type, its bridge index or the double coset. A plat presentation of a split link is split if the planar projection of the plat presentation is not connected. We prove that pocket moves are the only obstruction to representing split links by split plat presentations. Since any pocket move corresponds to a sequence of double coset moves, we have the corollary that the double coset of every plat presentation of a split link has a split plat presentation. We obtain an analogous result for composite links by utilizing flip moves, which were also first described in the second author's work, arXiv:2308.00732 [math.GT].
Paper Structure (14 sections, 10 theorems, 21 equations, 21 figures)

This paper contains 14 sections, 10 theorems, 21 equations, 21 figures.

Table of Contents

  1. Introduction
  2. Idea of proof
  3. Contrasting Plats with Closed Braids
  4. Outline of the paper
  5. Preliminaries
  6. System of Level Curves
  7. Moves on plats
  8. Braid isotopies
  9. Double coset moves and the Pocket move
  10. The flip move
  11. A special configuration of plats
  12. Sphere Foliation and Tiling
  13. Graph and complexity function
  14. Main Result

Key Result

Theorem 1

Let $\mathcal{L}$ be a split link type. Let $L$ be an $n$-bridge plat presentation of $\mathcal{L}$. Then, there exists a finite sequence of plat presentations of $\mathcal{L}$: such that $L_k$ is split and $L_{i+1}$ is obtained from $L_{i}$ via the following constant bridge index moves:

Figures (21)

  • Figure 1: A generic 2n-plat
  • Figure 2: Braid isotopy corresponding to Reidemeister II move
  • Figure 3: Braid isotopy corresponding to Reidemeister III move
  • Figure 4: Stabilizing and Destabilizing a Plat
  • Figure 5: Generators of $\mathcal{K}_4$
  • ...and 16 more figures

Theorems & Definitions (18)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Lemma 4
  • Definition 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 8 more