Crossing changes for pseudo-ribbon surface-knots
A surface-knot \(F \subset \mathbb{R}^4\), with \(F\) an oriented closed surface, is called pseudo-ribbon if its orthogonal projection \(\pi(F) \subset \mathbb{R}^3\) is an immersed surface whose singularities are disjoint transversal double curves. By adding crossing information along the double curves of \(\pi(F)\), we get a pseudo-ribbon diagram \(D\) for \(F\), which completely determines \(F\) up to isotopy. A crossing change along a double curve of \(D\) consists in inverting the relative crossing information. In the paper under review, the author addresses the question: is crossing change an unknotting operation for pseudo-ribbon diagrams of surface-knots? More explicitly, is it possible to transform any pseudo-ribbon diagram \(D\) of a surface-knot \(F\) by a sequence of crossing changes along double curves into a pseudo-ribbon diagram \(D'\) of a surface-unknot \(F'\)? Here, surface-unknot means the boundary of a 3-dimensional 1-handlebody embedded in \(\mathbb{R}^4\) [cf. \textit{F. Hosokawa} and \textit{A. Kawauchi}, Osaka J. Math. 16, 233--248 (1979; Zbl 0404.57020)]. Actually, the author shows that a sequence of crossing changes can be found in order to get \(\pi_1(\mathbb{R}^4 - F') \cong \mathbb Z\). It is still a conjecture whether this is equivalent to saying that \(F'\) is a surface-unknot. The proof is based on a combinatorial analysis of the Wirtinger presentation of \(\pi_1(\mathbb{R}^4 - F)\), in terms of a suitable graph-theoretical encoding of the relevant data concerning the crossings along the double curves of \(D\).
scientific article; zbMATH DE number 2147090