Diagram uniqueness for highly twisted knots

TL;DR

This paper proves that for certain highly twisted knots and links, the diagram with a fixed bridge number is unique under specific conditions, providing a canonical form for classification.

Contribution

It establishes a new uniqueness theorem for diagrams of knots and links with high twist regions and long length, based on bridge number enumeration.

Findings

Unique simple m-bridge diagrams exist under specified conditions.

Such diagrams serve as canonical forms for knot and link classification.

The result applies to infinitely many knots and links with high twist complexity.

Abstract

Frequently, knots are enumerated by their crossing number. However, the number of knots with crossing number $c$ grows exponentially with $c$, and to date computer-assisted proofs can only classify diagrams up to around twenty crossings. Instead, we consider diagrams enumerated by bridge number, following the lead of Schubert who classified 2-bridge knots in the 1950s. We prove a uniqueness result for this enumeration. Using recent developments in geometric topology, including distances in the curve complex and techniques with incompressible surfaces, we show that infinitely many knot and link diagrams have a unique simple $m$-bridge diagram. Precisely, if $m$ is at least three, if each twist region of the diagram has at least three crossings, and if the length $n$ of the diagram is sufficiently long, i.e., $n>4m(m-2)$, then such a diagram is unique up to obvious rotations. This projection gives a canonical form for such knots and links, and thus provides a classification of these knots or links.