Diagram uniqueness of even plats

TL;DR

This paper extends previous results on the uniqueness of minimal width plat projections of knots to even plats, showing that under certain conditions, such projections are unique, thereby broadening the classification of knot diagrams.

Contribution

The paper introduces new arguments to prove the uniqueness of minimal width plat projections for even plats with specific crossing and length conditions, expanding prior work on odd plats.

Findings

Uniqueness of minimal width plat projections for even plats under certain conditions.

Conditions include at least three crossings per twist region and sufficient length relative to width.

Doubles the class of knots with uniquely classifiable diagrams.

Abstract

Every knot has a plat projection, obtained by closing up a braid with bridges. The plat projection is determined by the number of strands and the number of rows of twist regions in the braid, and an integer number of crossings in each twist region. In recent work, we showed that under certain restrictions, including that the number of rows is odd, a minimal width plat projection is unique. In this paper we extend the results to even plats. Using new arguments, we show that if each of their twist regions contains at least three crossings, and their length is sufficiently long with respect to their width, then the projection is unique. This essentially "doubles" the set of knots for which such diagrams classify the links.