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 (σ -1 1 σ -1 2 ··· σ -1 p-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).