Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Braid positive surgery descriptions

Marc Kegel, Paula Truöl

Abstract

In this short note, we prove that every closed, oriented, connected 3-manifold arises as Dehn surgery along a braid positive link.

Braid positive surgery descriptions

Abstract

In this short note, we prove that every closed, oriented, connected 3-manifold arises as Dehn surgery along a braid positive link.

Paper Structure

This paper contains 4 sections, 3 theorems, 5 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Surgery diagrams and Rolfsen twists
  3. Garside theory
  4. Braid positive surgery diagrams

Key Result

Theorem 2

Every closed, oriented, connected $3$-manifold $M$ can be obtained by Dehn surgery along a braid positive link $L$ in $S^3$. Moreover, $L$ can be chosen to have at most $s(M) + 1$ many components.

Figures (4)

  • Figure 1: An $N$-fold Rolfsen twist.
  • Figure 2: A braid (left) and its closure (right).
  • Figure 3: The first row shows that a full twist $T_k$ commutes with any other braid $\beta$ on $k$ strands. The second row visualizes that a full twist $T_k$ followed by a negative Artin generator $\sigma_i^{-1}$ yields a positive braid.
  • Figure 4: Transforming a braid into a positive braid by a single surgery.

Theorems & Definitions (7)

  • Theorem 2
  • Lemma 3: Rolfsen twist rolfsen:rational_calc
  • proof
  • Lemma 4: GARSIDE
  • proof
  • proof : Proof of Theorem \ref{['thm:main']}
  • Remark 5