On the Non-Orientable $3$- and $4$-Genera of a Knot: Connections and Comparisons

Abstract: We define a new quantity, the Euler-normalized non-orientable genus, to connect a variety of ideas in the theory of non-orientable surfaces bounded by knots. This quantity is used to reframe non-orientable slice-torus bounds on the non-orientable $4$-genus, to bound below the Turaev genus as a measure of distance to an alternating knot, and to understand gaps between the $3$- and $4$-dimensional non-orientable genera of pretzel knots. Further, we make connections to essential surfaces in knot complements and the Slope Conjecture.