On a Nonorientable Analogue of the Milnor Conjecture

Abstract

The nonorientable 4-genus ${\gamma }_4(K)$ of a knot $K$ is the smallest first Betti number of any nonorientable surface properly embedded in the 4-ball, and bounding the knot $K$. We study a conjecture proposed by Batson about the value of ${\gamma }_4$ for torus knots, which can be seen as a nonorientable analogue of Milnor's Conjecture for the orientable 4-genus of torus knots. We prove the conjecture for many infinite families of torus knots, by relying on a lower bound for ${\gamma }_4$ formulated by Ozsváth, Stipsicz, and Szabó. As a side product we obtain new closed formulas for the signature of torus knots.

The article has been updated to account for the recent discovery by Andrew Lobb of a counterexample to Batson's Conjecture arXiv:1906.00799. The portion of the article comparing the nonorientable 3- and 4-genus of torus knots has been split off as a separate paper that will appear on the ArXiv shortly

References in corpus (2)

• On cobordisms between knots, braid index, and the Upsilon-invariant
• The non-orientable 4-genus for knots with 8 or 9 crossings

Cited by in corpus (5)

• Chern-Simons functional, singular instantons, and the four-dimensional clasp number
• Concordance invariants from the $E(-1)$ spectral sequence on Khovanov homology
• An infinite family of counterexamples to Batson's conjecture
• Nonorientable surfaces bounded by knots: a geography problem
• On the nonorientable four-ball genus of torus knots