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.