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 $p$.

arXiv comment31 pages, 8 figures. Comments are welcome! v2: Corresponds to version accepted for publication in Algebraic & Geometric Topology

References in corpus (18)

• Cobordisms of sutured manifolds and the functoriality of link Floer homology
• Involutive Heegaard Floer homology
• Heegaard Floer homology and rational cuspidal curves
• Lens space surgeries and a conjecture of Goda and Teragaito
• Link cobordisms and functoriality in link Floer homology
• Connected sums and involutive knot Floer homology
• A note on cobordisms of algebraic knots
• On cobordisms between knots, braid index, and the Upsilon-invariant
• The nonorientable four-genus of knots
• Chern-Simons functional, singular instantons, and the four-dimensional clasp number
• On the Upsilon invariant and satellite knots
• The non-orientable 4-genus for knots with 8 or 9 crossings
• On a Nonorientable Analogue of the Milnor Conjecture
• Concordance invariants from the $E(-1)$ spectral sequence on Khovanov homology
• Involutive Heegaard Floer homology and rational cuspidal curves
• An involutive upsilon knot invariant
• Non-orientable link cobordisms and torsion order in Floer homologies
• An infinite family of counterexamples to Batson's conjecture

Cited by in corpus (1)

• Non-orientable 4-genus of torus knots