Rigidity of highly twisted plat diagrams

TL;DR

This paper proves that highly twisted plat projections of knots and links are unique and canonical, providing a new classification method for such links based on their minimal plat representations.

Contribution

It establishes the conditions under which a plat projection is unique, introducing a canonical form for knots and links with complex plat diagrams.

Findings

Unique plat projections for knots with at least four crossings per twist region

Provides a canonical form for certain knots and links

Classifies links based on their minimal plat projections

Abstract

In this paper we prove that if a knot or link has a sufficiently complicated plat projection, then that plat projection is unique. More precisely, if a knot or link has a $2m$-plat projection, where $m$ is at least four, and height at least two, and each twist region of the plat contains at least four crossings, then such a projection is unique up to obvious rotations. In particular, this projection gives a canonical form for such knots and links, and thus provides a classification of these links.