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Shear coordinates and braid invariants

Vassily Olegovich Manturov

TL;DR

The paper develops braid invariants by embedding braids into hyperbolic geometry through shear coordinates on Voronoi/Delaunay structures. It assigns edge labels to the initial Delaunay triangulation and propagates them under flips via the Ptolemy transformation and shear coordinate updates, yielding Laurent polynomial coordinates in the initial data. The central result proves that isotopic braids induce identical label transformations, by leveraging pentagon relations and far commutativity, ensuring the constructed map T(β) is an invariant. This approach connects braid invariants with cluster-algebraic dynamics and offers a tropical analogue, expanding tools for studying braids via hyperbolic and combinatorial geometry.

Abstract

We present a way of using shear coordinates in hyperbolic geometry to get invariants of braids. This method also has a tropical analogue.

Shear coordinates and braid invariants

TL;DR

The paper develops braid invariants by embedding braids into hyperbolic geometry through shear coordinates on Voronoi/Delaunay structures. It assigns edge labels to the initial Delaunay triangulation and propagates them under flips via the Ptolemy transformation and shear coordinate updates, yielding Laurent polynomial coordinates in the initial data. The central result proves that isotopic braids induce identical label transformations, by leveraging pentagon relations and far commutativity, ensuring the constructed map T(β) is an invariant. This approach connects braid invariants with cluster-algebraic dynamics and offers a tropical analogue, expanding tools for studying braids via hyperbolic and combinatorial geometry.

Abstract

We present a way of using shear coordinates in hyperbolic geometry to get invariants of braids. This method also has a tropical analogue.
Paper Structure (5 sections, 1 theorem, 1 equation, 3 figures)

This paper contains 5 sections, 1 theorem, 1 equation, 3 figures.

Table of Contents

  1. Introduction
  2. From braids to Voronoi diagrams
  3. Acknowledgements
  4. The construction of the main invariant
  5. The main theorem

Key Result

Theorem 1

For two isotopic generic braids $\beta$ and $\beta'$ the transformations $T(\beta)$ and $T(\beta')$ are identical.

Figures (3)

  • Figure 1: The shear transformation of labels
  • Figure 2: The Ptolemy transformation of labels
  • Figure 4: The pentagon transformation

Theorems & Definitions (2)

  • Theorem
  • proof