Hopf Arborescent Links, Minor Theory, and Decidability of the Genus Defect

TL;DR

The paper addresses the decidability of four-dimensional genus defects for a structured class of knots and links called Hopf arborescent links, defined as boundaries of iterated Hopf-band plumbings encoded by plane trees. It introduces a tailored minor theory based on surface-minors and link-minors, and proves a well-quasi-ordering result via the Kruskal tree theorem, which in turn implies a monotonicity property for the genus defect $Δg = g - g_4$. Although the main decidability result is non-constructive, the approach shows that for any fixed $k$ there exists a finite excluded-minor family that certifies when $Δg leq k$, and outlines how one would algorithmically test for this via comparisons of plane-tree encodings and link equivalence. The paper also provides concrete examples demonstrating nontrivial defects and discusses how defects can be made arbitrarily large by combining isolated patterns, highlighting implications for the landscape of 4D topology and suggesting directions toward explicit algorithmic methods within this constrained knot class.

Abstract

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.

Figures (14)

• Figure 1: Top: A positive Hopf band and a plumbing. Bottom: An Hopf arborescent link and an associated planar tree.
• Figure 2: A negative Hopf band on the left and a positive one on the right with its core in red.
• Figure 3: A Hopf plumbing of a Hopf band $H$ on top of a Seifert surface $\Sigma$ along $\alpha$.
• Figure 4: A 3D-view of a Hopf arborescent link and its construction from two different plane trees. The chosen orientation of the root of each tree is indicated on the coloured core of the matching Hopf band. The orientation of the plane is counter-clockwise.
• Figure 5: Orientation of the green core when its associated Hopf band (unsigned as it does not matter for the rule) is plumbed on top of the Hopf band with the red core.

Theorems & Definitions (33)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Theorem 2.2
• proof : Proof of Theorem \ref{['th_fibred_surface_minimise']}
• Theorem 2.3: Kruskal_tree_minornash-williams_kruskal
• Definition 3.1
• Proposition 3.2
• Lemma 3.3
• proof