On a refinement of the non-orientable 4-genus of Torus knots.

In formulating a non-orientable analogue of the Milnor Conjecture on the 4-genus of torus knots, Batson [Math. Res. Lett. 21 (2014), pp. 423–436] developed an elegant construction that produces a smooth non-orientable spanning surface in B^4 for a given torus knot in S^3. While Lobb [Math. Res. Lett. 26 (2019), pp. 1789] showed that Batson's surfaces do not always minimize the non-orientable 4-genus, we prove that they do minimize among surfaces that share their normal Euler number. We also determine the possible pairs of normal Euler number and first Betti number for non-orientable surfaces whose boundary lies in a class of torus knots for which Batson's surfaces are non-orientable 4-genus minimizers. [ABSTRACT FROM AUTHOR]