Fibred double torus knots which are band-sums of torus knots

Abstract

A double torus knot $K$ is a knot embedded in a Heegaard surface $H$ of genus 2, and $K$ is non-separating if $H \setminus K$ is connected. In this paper, we determine the genus of a non-separating double torus knot that is a band-connected sum of two torus knots. We build a bridge between an algebraic condition and a geometric requirement (Theorem 5.5), and prove that such a knot is fibred if (and only if) its Alexander polynomial is monic, i.e. the leading coefficient is $\pm 1$. We actually construct fibre surfaces, using T. Kobayashi's geometric characterization of a fibred knot in our family. Separating double torus knots are also discussed in the last section.