An old and challenging conjecture proposed by Fox in 1962 states that the coefficients of the Alexander polynomial of any alternating knot are trapezoidal. In other words, these coefficients, increase, stabilize then decrease in a symmetrical way. This curious behavior of the Alexander polynomial has been confirmed in several special cases of alternating knots. The purpose of this paper is to prove that the conjecture holds for some families of alternating knots of braid index 3.

中文翻译：

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福克斯关于封闭三辫子的梯形猜想

Fox 在 1962 年提出的一个古老且具有挑战性的猜想指出，任何交替结的亚历山大多项式的系数都是梯形的。换句话说，这些系数以对称的方式增加、稳定然后减少。亚历山大多项式的这种奇怪行为已在几个交替结的特殊情况下得到证实。本文的目的是证明该猜想对一些编织指数为 3 的交替结族成立。