Minimal braid representatives of quasipositive links

TL;DR

This paper proves that every quasipositive link has a minimal braid representative that is also quasipositive, and uses this to establish bounds on self-linking numbers and classify amphicheiral quasipositive links.

Contribution

It demonstrates the existence of quasipositive minimal braid representatives for all quasipositive links, partially resolving a question by Orevkov.

Findings

Every quasipositive link has a quasipositive minimal braid representative.

The maximal self-linking number is bounded below by the negative of the minimal braid index.

The only amphicheiral quasipositive links are unlinks.

Abstract

We show that every quasipositive link has a quasipositive minimal braid representative, partially resolving a question posed by Orevkov. These quasipositive minimal braids are used to show that the maximal self-linking number of a quasipositive link is bounded below by the negative of the minimal braid index, with equality if and only if the link is an unlink. This implies that the only amphicheiral quasipositive links are the unlinks, answering a question of Rudolph's.