On the nonorientable four-ball genus of torus knots

Abstract

The nonorientable four-ball genus of a knot $K$ in $S^3$ is the minimal first Betti number of nonorientable surfaces in $B^4$ bounded by $K$. By amalgamating ideas from involutive knot Floer homology and unoriented knot Floer homology, we give a new lower bound on the smooth nonorientable four-ball genus ${\gamma }_4$ of any knot. This bound is sharp for several families of torus knots, including $T_{4n,(2n\pm 1{)}^2}$ for even $n\ge 2$, a family Longo showed were counterexamples to Batson's conjecture. We also prove that, whenever $p$ is an even positive integer and $\frac{p}{2}$ is not a perfect square, the torus knot $T_{p,q}$ does not bound a locally flat Möbius band for almost all integers $q$ relatively prime to $p$.

31 pages, 8 figures. Comments are welcome! v2: Corresponds to version accepted for publication in Algebraic & Geometric Topology

References in corpus (3)

• Concordance invariants from the $E(-1)$ spectral sequence on Khovanov homology
• An involutive upsilon knot invariant
• An infinite family of counterexamples to Batson's conjecture

Cited by in corpus (1)

• Non-orientable 4-genus of torus knots