Bypasses for rectangular diagrams. Proof of Jones' conjecture and related questions

TL;DR

This paper introduces criteria for simplifying rectangular diagrams via Legendrian knots, proves Jones' conjecture on braid invariants, and offers new proofs for existing theorems, advancing understanding of knot simplifications and invariants.

Contribution

It provides a new criterion for diagram simplification, proves Jones' conjecture, and offers a novel proof of the monotonic unknot simplification theorem.

Findings

A criterion for rectangular diagram simplification based on Legendrian knots.

Proof of Jones' conjecture on braid intersection invariance.

A new proof of the monotonic unknot simplification theorem.

Abstract

In the present paper a criteria for a rectangular diagram to admit a simplification is given in terms of Legendrian knots. It is shown that there are two types of simplifications which are mutually independent in a sense. A new proof of the monotonic simplification theorem for the unknot is given. It is shown that a minimal rectangular diagram maximizes the Thurston--Bennequin number for the corresponding Legendrian links. Jones' conjecture about the invariance of the algebraic number of intersections of a minimal braid representing a fixed link type is proved.