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Crosscap number: Handcuff graphs and unknotting number
Many types of invariants are used in the study of knots. Some are based on polynomials, some are purely algebraic, and some have their origins in geometry. One of the best known geometric knot invariants is the genus of a knot. A closely related but lesser-known invariant, crosscap number, was first introduced by Bradd Evans Clark in 1978. This thesis primarily concerns crosscap number two knots. Starting with a list of knots found to have crosscap number two by Burton and Ozlen using a linear programing approach, we verify, though are unable to expand, this list using a computer search. Surfaces that realize the minimal crosscap number of these knots are moreover found to arise from low-complexity handcuff diagrams. We also find a knot for which a single crossing change simultaneously lowers the unknotting number and raises the crosscap number. The proof utilizes signature to bound unknotting number from below. This result is a non-orientable analogue of a result for genus given in a paper by Scharlemann and Thompson. The result is further expanded to two infinite families of knots, one non-hyperbolic and one hyperbolic, which have the same property.^
Kerian, Anne, "Crosscap number: Handcuff graphs and unknotting number" (2015). ETD collection for University of Nebraska - Lincoln. AAI3716437.