# An algorithm to compute the crosscap number of a knot

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Jonathan Spreer

created by collari on 18 Oct 2023

modified on 20 Nov 2023

27 nov 2023
-- 09:30

**Abstract.**

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.