The nonorientable 4-genus is an invariant of knots which has been studied by many authors, including Gilmer and Livingston, Batson, and Ozsváth, Stipsicz, and Szabó. Given a nonorientable surface F in B⁴ with ∂ F = K in S³ a knot, an analysis of the existing methods for bounding and computing the nonorientable 4-genus reveals relationships between the first Betti number β₁ of F and the normal Euler class e of F. This relationship yields a geography problem: given a knot K, what is the set of realizable pairs (e(F), β₁(F)) where F in B⁴ is a nonorientable surface bounded by K? We explore this problem for families of torus knots. In addition, we use the Ozsváth-Szabó d-invariant of two-fold branched covers to give finer information on the geography problem. We present an infinite family of knots where this information provides an improvement upon the bound given by Ozsváth, Stipsicz, and Szabó using the Upsilon invariant.

Thanks are due to Charles Livingston for guidance and careful reading of many early versions of this paper. In addition, Ina Petkova provided many helpful comments and suggestions for improving the exposition.