Resolution depth of positive braids

TL;DR

This paper establishes a precise formula for the resolution depth of positive braid closures, linking it to the length of the braid word and the number of distinct generators.

Contribution

It provides a novel exact calculation for the resolution depth of positive braid closures based on braid word properties.

Findings

Resolution depth equals braid length minus number of distinct letters.

The result applies specifically to strictly positive braid words.

This advances understanding of link complexity in braid theory.

Abstract

The depth of a link measures the minimum height of a resolving tree for the link whose leaves are all unlinks. We show that the depth of the closure of a strictly positive braid word is the length of the word minus the number of distinct letters.