On a Nonorientable Analogue of the Milnor Conjecture

Stanislav Jabuka; Cornelia A. Van Cott

doi:10.2140/agt.2021.21.2571

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arXiv · 1809.01779

On a Nonorientable Analogue of the Milnor Conjecture

Stanislav Jabuka·Cornelia A. Van Cott

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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 Ozsv\'ath, Stipsicz, and Szab\'o. As a side product we obtain new closed formulas for the signature of torus knots.

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BibTeX RIS

Stanislav Jabuka, Cornelia A. Van Cott. 2018-09-06. On a Nonorientable Analogue of the Milnor Conjecture. https://doi.org/10.2140/agt.2021.21.2571

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