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：虽然结的亏格计算问题目前已相对清晰，但无论是光滑范畴还是拓扑局部平坦范畴，其四维变体的算法仍然未知。本文研究一类称为Hopf树状链接的结与链接，它们是由Hopf带的若干次迭代拼接的边界所构成。我们证明，对于此类链接，度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，其关键在于证明：在由其Seifert曲面包含关系定义的子式关系下，Hopf树状链接的Seifert曲面形成一个良拟序。