Untangling trigonal diagrams

TL;DR

This paper demonstrates a method to transform trigonal diagrams of certain links into alternating forms without increasing crossings, maintaining trigonal structure throughout the process.

Contribution

It introduces a procedure to convert trigonal diagrams of specific links into alternating diagrams while preserving the trigonal form and not increasing crossings.

Findings

Transformation to alternating diagrams is always possible for the given links.

The process preserves the trigonal structure at each step.

Crossing number does not increase during transformation.

Abstract

Let $K$ be a link of Conway's normal form $C(m)$, $m \geq 0$, or $C(m,n)$ with $mn\textgreater{}0$, and let $D$ be a trigonal diagram of $K.$ We show that it is possible to transform $D$ into an alternating trigonal diagram, so that all intermediate diagrams remain trigonal, and the number of crossings never increases.