Divide knots of maximal genus defect

We construct divide knots with arbitrary smooth four-genus but topological four-genus equal to one. In particular, for strongly quasipositive fibred knots, the ratio between the topological and the smooth four-genus can be arbitrarily close to zero.

💡 Research Summary

This paper, titled “Divide knots of maximal genus defect,” constructs and analyzes a family of knots that exhibit an extreme disparity between two important four-dimensional invariants: the smooth four-genus and the topological four-genus.

The central result (Theorem 1) proves that for every positive integer g, there exists a “divide knot” whose smooth four-genus is g, but whose topological four-genus is exactly one. Divide knots are a class of knots introduced by A’Campo, arising from immersed curves (“divides”) in a disc and lying within the unit sphere bundle of the disc’s tangent bundle (diffeomorphic to S³). This class is contained within the broader class of strongly quasipositive fibred knots.

The smooth four-genus of a knot, denoted g₄, is the minimal genus of a smoothly embedded, orientable surface bounding the knot in the standard smooth 4-ball. The topological four-genus, denoted g₄top, is the analogous minimal genus but for surfaces that are only required to be locally flat and topologically embedded. For many knots, these numbers coincide, but they can differ. A key known fact, by Rudolph’s generalization of the Thom conjecture, is that for strongly quasipositive knots (including divide knots), the smooth four-genus equals the classical Seifert genus.

The authors’ explicit examples are the knots Kₙ obtained from the “snail divide” with n double points. By the above fact, g₄(Kₙ) equals its Seifert genus, which is n. The main technical achievement is proving that g₄top(Kₙ) = 1 for all n.

The proof proceeds as follows:

1. The canonical, genus-minimizing Seifert surface Σ for Kₙ is described via Ishikawa’s plumbing construction of positive Hopf bands. A specific basis for its first homology H₁(Σ;ℤ) is given, consisting of curves associated to the inner regions (αᵢ) and double points (γᵢ) of the divide.
2. Lemma 4 provides a complete calculation of the Seifert form S on this basis, using geometric arguments about linking numbers and the checkerboard coloring of the divide complement.
3. Proposition 5 constructs a subgroup V of H₁(Σ;ℤ) of rank 2n-2, generated by elements aᵢ = αᵢ₊₁ - γᵢ and bᵢ = γᵢ₊₁. Through a detailed computation using Lemma 4, it is shown that for a matrix A representing the Seifert form restricted to V, the Laurent polynomial det(tA - Aᵀ) is a unit in ℤ