Virtual Knot Theory --Unsolved Problems

For any two links in the $3$-sphere, we provide a visual construction of a stable map $f$ from the $3$-sphere to the Euclidean plane such that $f$ has no cusp points, the set of definite (resp. indefinite) fold points of $f$ is isotopic to the first (resp. second) given link, and $f$ does not have a certain type of singular fibers. Moreover, we obtain a similar result for any $3$-manifold by setting the target space to the $2$-sphere.

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arXiv2026

The Smoothability of Embedded Surfaces in 4-Manifolds Under Stabilizations

We study the smooth stabilization number of a locally flat embedded surface in a smooth $4$-manifold, defined as the minimum number of connected sums with $S^2\times S^2$ needed to make the surface topologically isotopic to a smooth embedding. We prove that a nonsmoothable locally flat surface in a closed, simply connected smooth $4$-manifold becomes smoothable after a single stabilization if its homology class is primitive and noncharacteristic and its complement has abelian fundamental group. This yields infinitely many examples of nonsmoothable surfaces with smooth stabilization number one. We also exhibit examples for which a single stabilization produces infinitely many pairwise smoothly nonisotopic representatives in the same topological isotopy class. In contrast, we construct a locally flat genus-eight surface representing $6h$ in $\mathbb{CP}^2$ whose smooth stabilization number is exactly two. More generally, we show that smooth stabilization numbers of locally flat embedded surfaces in $\mathbb{CP}^2$ are unbounded.

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arXiv2026

Sections, pseudosections, and multisections of Lefschetz fibrations

We show that not every symplectic Lefschetz fibration over the $2$--sphere admits a smooth section, settling a long-standing open problem. To prove this, we develop a method for studying sections via lifts of monodromy factorizations with point-pushing maps and the induced handle decomposition of the total space, and apply it to an infinite family of Lefschetz fibrations. In contrast, we show that every Lefschetz fibration over $S^2$ admits positive multisections. Some of these examples admit no multisections with spherical components. We also construct signature-zero Lefschetz fibrations of every odd genus $g\geq 7$, which in turn give symplectic Lefschetz fibrations of any prescribed signature in each of these genera.

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