On iterated torus knots and transversal knots

TL;DR

This paper proves that iterated torus knots are exchange reducible, which implies they are transversally simple, thereby advancing understanding of their classification in contact topology.

Contribution

It establishes that all iterated torus knots are exchange reducible, leading to their classification as transversally simple in contact topology.

Findings

Iterated torus knots are exchange reducible.

Exchange reducibility implies transversally simplicity.

Iterated torus knots are transversally simple.

Abstract

A knot type is exchange reducible if an arbitrary closed n-braid representative can be changed to a closed braid of minimum braid index by a finite sequence of braid isotopies, exchange moves and +/- destabilizations. In the manuscript [J Birman and NC Wrinkle, On transversally simple knots, preprint (1999)] a transversal knot in the standard contact structure for S^3 is defined to be transversally simple if it is characterized up to transversal isotopy by its topological knot type and its self-linking number. Theorem 2 of Birman and Wrinkle [op cit] establishes that exchange reducibility implies transversally simplicity. The main result in this note, establishes that iterated torus knots are exchange reducible. It then follows as a Corollary that iterated torus knots are transversally simple.