On a Nonorientable Analogue of the Milnor Conjecture
Abstract
The nonorientable 4-genus of a knot is the smallest first Betti number of any nonorientable surface properly embedded in the 4-ball, and bounding the knot . We study a conjecture proposed by Batson about the value of 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 formulated by Ozsv\'ath, Stipsicz, and Szab\'o. As a side product we obtain new closed formulas for the signature of torus knots.
Cite
@article{arxiv.1809.01779,
title = {On a Nonorientable Analogue of the Milnor Conjecture},
author = {Stanislav Jabuka and Cornelia A. Van Cott},
journal= {arXiv preprint arXiv:1809.01779},
year = {2021}
}
Comments
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