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Title: Log-Concavity of the Alexander Polynomial
Abstract The central question of knot theory is that of distinguishing links up to isotopy. The first polynomial invariant of links devised to help answer this question was the Alexander polynomial (1928). Almost a century after its introduction, it still presents us with tantalizing questions, such as Fox’s conjecture (1962) that the absolute values of the coefficients of the Alexander polynomial $$\Delta _{L}(t)$$ of an alternating link $$L$$ are unimodal. Fox’s conjecture remains open in general with special cases settled by Hartley (1979) for two-bridge knots, by Murasugi (1985) for a family of alternating algebraic links, and by Ozsváth and Szabó (2003) for the case of genus $$2$$ alternating knots, among others. We settle Fox’s conjecture for special alternating links. We do so by proving that a certain multivariate generalization of the Alexander polynomial of special alternating links is Lorentzian. As a consequence, we obtain that the absolute values of the coefficients of $$\Delta _{L}(t)$$, where $$L$$ is a special alternating link, form a log-concave sequence with no internal zeros. In particular, they are unimodal.  more » « less
Award ID(s):
1847284
PAR ID:
10633590
Author(s) / Creator(s):
; ;
Publisher / Repository:
Oxford Academic
Date Published:
Journal Name:
International Mathematics Research Notices
Volume:
2024
Issue:
13
ISSN:
1073-7928
Page Range / eLocation ID:
10273 to 10284
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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