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.

中文翻译：

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亚历山大多项式的对数凹性

纽结理论的核心问题是区分同位素的链接。旨在帮助回答这个问题的第一个链接多项式不变量是亚历山大多项式（1928）。在它提出近一个世纪后，它仍然向我们提出了诱人的问题，例如福克斯的猜想（1962），交替链路 $L 的亚历山大多项式 $\Delta _{L}(t)$ 的系数的绝对值$ 是单峰的。 Fox 的猜想总体上仍然是开放的，Hartley (1979) 解决了两桥结的特殊情况，Murasugi (1985) 解决了交替代数链族的问题，Ozsváth 和 Szabó (2​​003) 解决了 $2$ 交替属的情况结等。我们解决了福克斯关于特殊交替链接的猜想。我们通过证明特殊交替链接的亚历山大多项式的某个多元推广是洛伦兹式来实现这一点。因此，我们得到 $\Delta _{L}(t)$ 的系数的绝对值，其中 $L$ 是一个特殊的交替链接，形成一个没有内部零的对数凹序列。特别是，它们是单峰的。