22 oct 2024 -- 14:30
Aula Riunioni, Dipartimento di Matematica, Pisa
Seminario di Geometria
Abstract.
In this talk I will present some results on how primeness of fibered links behaves under an operation called the Murasugi sum, which generalises the connected sum. As an application, this allows us to resolve a 30 year old conjecture of Cromwell in the case of braid closures. Cromwell showed that, if a link diagram represented as a positive braid features no circles that decompose it as a connected sum (in other words, it is not "obviously" composite), then the link is indeed prime. In his words, "positive braids are visually prime". He further conjectured that the same would hold for link diagrams that yield minimal genus Seifert surfaces when Seifert's algorithm is applied. If the diagram is required to be a braid closure, then Cromwell's condition is equivalent to the braid being homogeneous. Thus, we show that homogeneous braids are visually prime. This is joint work with Peter Feller and Lukas Lewark.