Fundamental Groups, 3-Braids, and Effective Estimates of Invariants

TL;DR

This paper introduces new invariants for pure three-braids and provides effective bounds for these invariants and their entropy, using a natural syllable decomposition and fundamental group analysis.

Contribution

It defines braid invariants based on syllable decomposition and establishes effective bounds for these invariants and entropy in pure three-braids.

Findings

Effective upper and lower bounds for braid invariants.

Bounds differ by a constant factor independent of the word.

Bounds for entropy of pure three-braids.

Abstract

We define invariants of braids rather than invariants of conjugacy classes of braids. For any pure three-braid we give effective upper and lower bounds for these invariants. This is done in terms of a natural syllable decomposition of the word representing the image of the braid in the braid group modulo its center. The bounds differ by a multiplicative constant not depending on the word. Respective bounds are given for all three-braids. We also obtain effective upper and lower bounds for the entropy of pure three-braids in these terms. The proof leads to the study of the extremal length of classes of curves representing elements of the fundamental group of the twice punctured complex plane.