27 nov 2023 -- 09:30
The crosscap number of a knot is the non-orientable counterpart of its genus. It is defined as the minimum of one minus the Euler characteristic of S, taken over all non-orientable surfaces S bounding the knot. Computing the crosscap number of a knot is tricky, since normal surface theory - the usual tool to prove computability of problems in 3-manifold topology, does not deliver the answer "out-of-the-box". In this talk, I will review the strengths and weaknesses of normal surface theory, focusing on why we need to work to obtain an algorithm to compute the crosscap number. I will then explain the theorem stating that an algorithm due to Burton and Ozlen can be used to give us the answer. This is joint work with Jaco, Rubinstein, and Tillmann.