By Stefan Friedl

We show that two hypersurfaces in a manifold are related by a sequence of embedded cobordisms if and only if they represent the same homology class. By applying handle decompositions we turn these cobordisms into a sequence of embedded surgeries. Specializing to Seifert surfaces we obtain a conceptual proof that two Seifert surfaces of a fixed link are related by tube attachments and tube removals.

This is joint work with Tobias Hirsch, Clayton McDonald, José Pedro Quintanilha and Daniel Zach.

By Maciej Borodzik

A strongly invertible knot is a knot K in S³ that is invariant under an involution Z₂ acting on S³ and such that this involution has two
fixed points on K.

Strongly invertible knots are a rich source of example explaining exotic behavior of surfaces in B⁴. For example, there exist strongly invertible knots that are homotopy slice, but not slice.

In the talk, I will present recent developments on strongly invertible knots, focusing on the results by Dai, Mallick, Stoffregen and myself.

By Eva Horvat

The fundamental quandle is a topological invariant of links in Euclidean spaces. We describe the fundamental quandle of a properly embedded surface (possibly with boundary) in the product of the Euclidean space with an interval. We derive its presentation in terms of a motion picture diagram or a CH-diagram of the surface. We prove that a ribbon concordance C from a classical knot K₁ to K₀ gives rise to an injective quandle homomorphism Q(K₀) → Q(C) and a surjective quandle homomorphism Q(K₁) → Q(C).

By Wojciech Politarczyk

The annular Rasmussen invariant, introduced and studied by Grigsby, Licata, and Wehrli, generalizes the classical Rasmussen ‭s-invariant. In this talk, we discuss the application of this invariant to computing the splitting numbers of links. Recall that the splitting number measures how far a link deviates from being a split sum of knots.

By Joseph Przytycki

Skein modules, from their humble beginning in April 1987, grew into a mainstream branch of mathematics. I will outline the path Skein Modules took, from my personal historical perspective.

By Boštjan Gabrovšek

Lasso motifs are complex, entangled protein structures where a covalently closed loop is pierced by one or both termini of the polypeptide chain. We propose a new method for the detection and classification of lasso proteins using persistent homology (PH). By analyzing the interaction between the protein loop and tail through changes in persistence diagrams, we identify piercing events as tail atoms are progressively incorporated into the loop structure. This method uses bottleneck distances to detect transitions in the protein’s topological type, specifically, the number of local maxima in the bottleneck distance diagram corresponds to the number of piercings.

Our approach was validated against 4,846 proteins from the LassoProt and AlphaLasso database, showing strong agreement with existing minimal surface methods while offering improved computational efficiency and stability.

By Lisa Piccirillo

Symplectic manifolds provide an important testing ground for the smooth 4-manifold topologist; unlike their smooth counterparts, symplectic manifolds are known to satisfy some amount of structural rigidity. In this talk, we introduce a class of not-necessarily symplectic smooth 4-manifolds which admit some rigidity properties analogous to those which arise in the symplectic setting.

This is joint work with Tye Lidman. 

By Daniel Kasprowski

Building on previous work on the homotopy classification of 4-manifolds with boundary, we give necessary and sufficient conditions for locally flat surfaces in simply-connected 4-manifolds to be equivalent.

This is joint work with Anthony Conway.