Knots and Surfaces in four-manifolds

Smooth four-manifolds – 4D topological manifolds with a smooth structure – are not well characterised. Funded by the European Research Council, the KnotSurf4d project aims to address this gap in the knowledge of high-dimensional manifolds by leveraging the genus function and its enhanced version that takes knots and their slice surfaces into account. This novel approach, which emphasises knots and their slice properties in various four-manifolds, could ultimately provide a candidate for an invariant that is a smooth generalisation of the intersection form and characterises smooth four-manifolds. The team will also study divisibility and torsion questions in the concordance group via knot Floer homology, as well as potential counterexamples for the famous Slice-Ribbon conjecture.

Publications

1. M. Beke, O. Plamenevskaya, L. Starkston. An Unexpected Rational Blowdown. arXiv:2510.00115
2. M. Eper, Sz. Szabo. Rank three representations of Painleve systems: III. Dolbeault structure, spectral correspondence. arXiv:2509.00418
3. M. Eper, Sz. Szabo. Rank three representations of Painlevé systems: I. Wild character varieties. arXiv:2505.21186
4. M. Beke. Minimal rational graphs admitting a QHD smoothing. arXiv:2504.06929
5. J. Hanselman, M. Marengon, B. Wong. Correction terms of double-branched covers and symmetries of immersed curves. arXiv:2408.02857
6. M. Eper, Sz. Szabo. Asymptotic geometry of non-abelian Hodge theory and Riemann--Hilbert correspondence, rank three E˜6 case. arXiv:2407.20638
7. R. I. Baykur, A. I. Stipsicz, Z. Szabo. Smooth structures on four-manifolds with finite cyclic fundamental groups. arXiv:2406.09007
8. Sz. Szabó, F. Živanović. Floer-theoretic filtration on Painlevé Hitchin systems. arXiv:2405.14850

Conferences

1. Deformation of Complex Singularities and Related Topics 2025.01.12-17.

Invited Speakers

1. Aceto Paolo
2. Santoro Diego
3. Hedden Matthew
4. Stefan Friedl
5. Brendan Owens
6. Gironella Fabio
7. Antonio Alfieri