Derivation of Schubert normal forms of 2-bridge knots from (1,1)-diagrams

Abstract

In this paper, we show that the dual [Formula: see text]-diagram of a [Formula: see text]-diagram (a.k.a. a two-pointed genus one Heegaard diagram) [Formula: see text] with [Formula: see text] and [Formula: see text] is given by [Formula: see text] where [Formula: see text] is the multiplicative inverse of [Formula: see text] modulo [Formula: see text] with [Formula: see text]. We also present explicitly how to derive a Schubert normal form of a [Formula: see text]-bridge knot from the dual [Formula: see text]-diagram of [Formula: see text] using weakly K-reducibility of [Formula: see text]-decompositions. This gives an alternative proof of Grasselli and Mulazzani’s result asserting that [Formula: see text] is a [Formula: see text]-diagram of the [Formula: see text]-bridge knot with a Schubert normal form [Formula: see text].