Embedded annuli and Jones' conjecture

TL;DR

This paper proves a key conjecture linking braid index and algebraic length by demonstrating the existence of embedded annuli between braids of the same link type after certain stabilizations and isotopies.

Contribution

It introduces a new method involving stabilizations and isotopies to establish the generalized Jones conjecture for braids within a link type.

Findings

Embedded annuli exist between braids after stabilizations and isotopy.

The generalized Jones conjecture is proven for all links.

A new approach to relating braid invariants is developed.

Abstract

We show that after stabilizations of opposite parity and braid isotopy, any two braids in the same topological link type cobound embedded annuli. We use this to prove the generalized Jones conjecture relating the braid index and algebraic length of closed braids within a link type, following a reformulation of the problem by Kawamuro.