Using a result of Takata, we prove a formula for the colored Jones polynomial of the double twist knots K(−m,−p) and K(−m,p) where m and p are positive integers. In the (−m,−p) case, this leads to new families of q-hypergeometric series generalizing the Kontsevich–Zagier series. Comparing with the cyclotomic expansion of the colored Jones polynomials of K(m,p) gives a generalization of a duality at roots of unity between the Kontsevich–Zagier function and the generating function for strongly unimodal sequences.

中文翻译：

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双绞结的彩色琼斯多项式和 Kontsevich-Zagier 级数

利用高田的结果，我们证明了双绞结的彩色琼斯多项式的公式ķ(-米,-p)和ķ(-米,p)在哪里米和p是正整数。在里面(-米,-p)情况下，这会导致新的家庭q- 泛化 Kontsevich-Zagier 级数的超几何级数。与有色琼斯多项式的分圆展开比较ķ(米,p)给出了 Kontsevich-Zagier 函数和强单峰序列的生成函数之间统一根的对偶性的概括。