Title: Knotted Surfaces at the End of the Rainbow

Abstract: There are numerous diagrammatic representations of knotted surfaces in four dimensions, including triplane diagrams, braid movies, banded unlinks, and braid charts. In this talk, I will introduce rainbow diagrams, a new representation of knotted surfaces in the 4-sphere that  bridges triplane diagrams and braid charts. By adapting classical methods from three-dimensional braid theory due to Alexander, Morton, and Yamada, we develop explicit procedures for converting between these diagrammatic representations. These methods also yield four-dimensional analogues of classical results from braid theory. This is joint work with Roman Aranda, Scott Carter, and Puttipong Pongtanapaisan.

Title: The Conway knot has infinite concordance order

Abstract: We examine how the Rasmussen invariant, satellite operations, and null-homologous twists can be used to establish infinite order of knots in the smooth concordance group. As an application, we show that the Conway knot has infinite concordance order. This is based on joint work with Chiara Donatone, Lukas Lewark, and Paula Truöl.

Title: Multisecting n-manifolds

Abstract: Multisections, as introduced by Ben Aribi--Courte--Golla--Moussard, generalize Heegaard splittings and trisections to arbitrary dimensions. Their existence, however, had only been established in dimension 3 (Heegaard), 4 (Gay--Kirby), and 5 (ACGM). We prove the existence of multisections in arbitrary dimensions.

One can also define a notion of bridge positions for submanifolds of codimension at least 2 in a multisection, generalizing bridge position for links in Heegaard splittings, bridge trisections for surfaces in trisections by Meier--Zupan, and bridge quadrisections in quadrisections by Aranda--Blackwell--Kim--Naylor--Pongtanapaisan. We give a proof that every submanifold of codimension at least 2 in a multisection can be put into bridge position, generalizing results in aforementioned work. This result is also a key ingredient in our proof of the existence of multisections. Time permitting, we illustrate our proof idea in some low-dimensional examples. We also mention some natural questions that arise from our work. This is joint work with Sylvain Courte, Delphine Moussard, and Xiaozhou Zhou.