The topological 4-genus and forbidden minor characterisations

Famously, a finite graph is planar exactly if it does not admit two specific graphs as minors. In fact, a characterisation by finitely many forbidden minors exists for any property which passes to minors with respect to a well-quasi-order. In this context, we discuss a minor relation for Seifert surfaces embedded in three-space, defined by isotopy into an incompressible subsurface, and the property to have an equality between the ordinary Seifert genus and the topological 4-genus of the boundary knot. In particular, we characterise this equality in the case of positive braid knots.