The minimal sequence of Reidemister moves bringing the diagram of $(n+1,n)$-torus knot to that of $(n,n+1)$-torus knot

TL;DR

This paper determines the minimal sequence of Reidemeister moves needed to transform the diagram of the $(n+1,n)$-torus knot into that of the $(n,n+1)$-torus knot, using cowrithe to prove minimality.

Contribution

It explicitly calculates the minimal number of Reidemeister moves required for the transformation and proves its minimality using cowrithe invariants.

Findings

The sequence consists of one RI move and $(n-1)n(2n-1)/6$ RIII moves.

The total number of moves is explicitly computed as $(n-1)n(2n-1)/6 + 1$.

The minimality of this sequence is established via cowrithe invariants.

Abstract

Let $D(p,q)$ be the usual knot diagram of the $(p,q)$-torus knot, that is, $D(p,q)$ is the closure of the $p$-braid $(\sigma_1^{-1} \sigma_2^{-1}... \sigma_{p-1}^{-1})^q$. As is well-known, $D(p,q)$ and $D(q,p)$ represent the same knot. It is shown that $D(n+1,n)$ can be deformed to $D(n,n+1)$ by a sequence of $\{(n-1)n(2n-1)/6 \} + 1$ Reidemeister moves, which consists of a single RI move and $(n-1)n(2n-1)/6$ RIII moves. Using cowrithe, we show that this sequence is minimal over all sequences which bring $D(n+1,n)$ to $D(n,n+1)$.