Semi-Adequate Closed Braids and Volume

TL;DR

This paper establishes bounds on the volumes of certain A-adequate closed braids using combinatorial and polynomial invariants, supporting the Coarse Volume Conjecture and linking algebraic parameters to geometric volume.

Contribution

It introduces new volume bounds for A-adequate closed braids based on combinatorial, polynomial, and normal form parameters, extending previous conjectures.

Findings

Volumes are bounded by twist number and braid string count.

Volumes relate to a stable coefficient of the colored Jones polynomial.

Parameters from Schreier normal form can be derived from the braid word.

Abstract

In this paper, we show that the volumes for a family of A-adequate closed braids can be bounded above and below in terms of the twist number, the number of braid strings, and a quantity that can be read from the combinatorics of a given closed braid diagram. We also show that the volumes for many of these closed braids can be bounded in terms of a single stable coefficient of the colored Jones polynomial, thus showing that this collection of closed braids satisfies a Coarse Volume Conjecture. By expanding to a wider family of closed braids, we also obtain volume bounds in terms of the number of positive and negative twist regions in a given closed braid diagram. Furthermore, for a family of A-adequate closed 3-braids, we show that the volumes can be bounded in terms of the parameter $s$ from the Schreier normal form of the 3-braid. Finally we show that, for the same family of A-adequate closed 3-braids, the parameters $k$ and $s$ from the Schreier normal form can actually be read off of the original 3-braid word.