Minimal sequences of Reidemeister moves on diagrams of torus knots

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 $(\sigma _1^{-1} \sigma _2^{-1} \cdots \sigma _{p-1}^{-1})^q$. 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 $D(n,n+1)$.