  We use composition of binary quadratic forms to systematically create pairs
of Seifert surfaces that are non-isotopic in the four-ball. Our main
topological result employs Gauss composition to classify the pairs of binary
quadratic forms that arise as the Seifert forms of pairs of disjoint Seifert
surfaces of genus one. The main ingredient of the proof is number-theoretic and
of independent interest. It establishes a new connection between the Bhargava
cube and the geometric approach to Gauss composition via planes in the space of
two-by-two matrices. In particular, we obtain a geometric recipe that given any
two binary quadratic forms finds a Bhargava cube that gives rise to their
composition.
