While the problem of computing the genus of a knot is now fairly well understood, no algorithm is known for its four-dimensional variants, both in the smooth and in the topological locally flat category. In this article, we investigate a class of knots and links called Hopf arborescent links, which are obtained as the boundaries of some iterated plumbings of Hopf bands. We show that for such links, computing the genus defects, which measure how much the four-dimensional genera differ from the classical genus, is decidable. Our proof is non-constructive, and is obtained by proving that Seifert surfaces of Hopf arborescent links under a relation of minors defined by containment of their Seifert surfaces form a well-quasi-order.

翻译：虽然计算纽结亏格的问题现已相当成熟，但在光滑范畴和拓扑局部平坦范畴中，其四维变体的算法仍未知。本文研究一类名为Hopf树状链环的纽结与链环，它们是由Hopf带的某些迭代拼接的边界得到的。我们证明，对于此类链环，衡量四维亏格与经典亏格差异的亏格缺陷是可判定的。我们的证明是非构造性的，通过证明Hopf树状链环的Seifert曲面在由包含关系定义的子式关系下构成一个良拟序来实现。