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

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 $D(n,n+1)$.

10pages, 17 figures

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