Algorithms in 4-manifold topology [submitted Nov 2024]: arXiv:2411.08775 | with S. Bastl, R. Burke, R. Chatterjee, S. Dey, A. Durst, S. Friedl, D. Galvin, A. García Rivas, T. Hirsch, C. Hobohm, C.-S. Hsueh, M. Kegel, F. Kern, S. Lee, C. Löh, N. Manikandan, L. Mousseau, L. Munser, M. Pencovitch, P. Perras, M. Powell, J. Quintanilha, L. Schambeck, D. Suchodoll, M. Tancer, A. Thiele, P. Truöl, M. Uschold, S. Veselá, M. Weiß, M. von Wunsch-Rolshoven.
Abstract: We show that there exists an algorithm that takes as input two closed, simply connected, topological 4-manifolds and decides whether or not these 4-manifolds are homeomorphic. In particular, we explain in detail how closed, simply connected, topological 4-manifolds can be naturally represented by a Kirby diagram consisting only of 2-handles. This representation is used as input for our algorithm. Along the way, we develop an algorithm to compute the Kirby-Siebenmann invariant of a closed, simply connected, topological 4-manifold from any of its Kirby diagrams and describe an algorithm that decides whether or not two intersection forms are isometric.
In a slightly different direction, we discuss the decidability of the stable classification of smooth manifolds with more general fundamental groups. Here we show that there exists an algorithm that takes as input two closed, oriented, smooth 4-manifolds with fundamental groups isomorphic to a finite group with cyclic Sylow 2-subgroup, an infinite cyclic group, or a group of geometric dimension at most 3 (in the latter case we additionally assume that the universal covers of both 4-manifolds are not spin), and decides whether or not these two 4-manifolds are orientation-preserving stably diffeomorphic.
On unknotting positive fibered knots and braids [to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci.]: arXiv:2312.07339 (Dec 2023, revised Sep 2024) | with M. Kegel, L. Lewark, N. Manikandan, F. Misev, and M. Silvero.
Abstract: The unknotting number u and the genus g of braid positive knots are equal, as shown by Rudolph. We prove the stronger statement that any positive braid diagram of a genus g knot contains g crossings, such that changing them produces a diagram of the trivial knot. Then, we turn to unknotting the more general class of fibered positive knots, for which u=g was conjectured by Stoimenow. We prove that the known ways to unknot braid positive knots do not generalize to fibered positive knots. Namely, we prove that there are fibered positive knots that cannot be unknotted optimally along fibered positive knots; there are fibered positive knots that do not arise as trefoil plumbings; and there are positive diagrams of fibered positive knots of genus g that do not contain g crossings, such that changing them produces a diagram of the trivial knot. In fact, we conjecture that one of our examples is a counterexample to Stoimenow's conjecture.
Khovanov homology of positive links and of L-space knots [submitted April 2023]: arXiv:2304.13613 | with M. Kegel, N. Manikandan, and M. Silvero.
Abstract: We determine the structure of the Khovanov homology groups in homological grading 1 of positive links. More concretely, we show that the first Khovanov homology is supported in a single quantum grading determined by the Seifert genus of the link, where the group is free abelian and of rank determined by the Seifert graph of any of its positive link diagrams. In particular, for a positive link, the first Khovanov homology is vanishing if and only if the link is fibered. Moreover, we extend these results to (p,q)-cables of positive knots whenever q≥p. We also show that several infinite families of Heegaard Floer L-space knots have vanishing first Khovanov homology. This suggests a possible extension of our results to L-space knots.
Knot invariants of the SnapPy census knots [ongoing work]: with K. Baker, J. Bohl, M. Kegel, D. McCoy, D. Suchodoll, and N. Weiss