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On eigenvalues of double branched covers


Author: Kouki Sato
Journal: Proc. Amer. Math. Soc. 147 (2019), 2707-2722
MSC (2010): Primary 57M25, 57M57
DOI: https://doi.org/10.1090/proc/14378
Published electronically: March 1, 2019
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Abstract: For a given knot, we study the minimal number of positive eigenvalues of the double branched cover over spanning surfaces for the knot. The value gives a lower bound for various genera, the dealternating number and the alternation number of knots, and we prove that Batson's bound for the non-orientable 4-genus gives an estimate of the value. In addition, we use the value to give a necessary condition for being quasi-alternating.


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Additional Information

Kouki Sato
Affiliation: Department of Mathematics, University of Tokyo, 7-1-2 Hongo Bunkyo-ku Tokyo, Japan 1130033
Email: sato.kouki@mail.u-tokyo.ac.jp

DOI: https://doi.org/10.1090/proc/14378
Received by editor(s): September 17, 2017
Received by editor(s) in revised form: June 19, 2018, and September 3, 2018
Published electronically: March 1, 2019
Additional Notes: The author was supported by JSPS KAKENHI Grant Number 15J10597.
Communicated by: David Futer
Article copyright: © Copyright 2019 American Mathematical Society