Begin with two knots and . Simon conjectured that if the knot group of surjects onto that of , then the genera of the orientable surfaces that the two knots bound are constrained. Specifically, he conjectured , where denotes the genus of . This conjecture has been proved for alternating knots and can be strengthened to an even stronger result in the case of two-bridge knots. In this talk, we consider the same sorts of questions, but in the world of nonorientable surfaces. We focus on two-bridge knots and find relationships among their crosscap numbers. This is joint work with Jim Hoste and Pat Shanahan.
References:
Hoste, J., Shanahan, P., Van Cott, C., “Crosscap number and the partial order on two-bridge knots”, https://arxiv.org/abs/2010.05009
