We obtain a formula for the Turaev-Viro invariants of a link complement in terms of values of the colored Jones polynomial of the link. As an application we give the first examples for which the volume conjecture of Chen and the third named author\,\cite{Chen-Yang} is verified. Namely, we show that the asymptotics of the Turaev-Viro invariants of the Figure-eight knot and the Borromean rings complement determine the corresponding hyperbolic volumes. Our calculations also exhibit new phenomena of asymptotic behavior of values of the colored Jones polynomials that seem not to be predicted by neither the Kashaev-Murakami-Murakami volume conjecture and various of its generalizations nor by Zagier's quantum modularity conjecture. We conjecture that the asymptotics of the Turaev-Viro invariants of any link complement determine the simplicial volume of the link, and verify it for all knots with zero simplicial volume. Finally we observe that our simplicial volume conjecture is stable under connect sum and split unions of links.

中文翻译：

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Turaev–Viro 不变量、彩色琼斯多项式和体积

我们根据链接的彩色琼斯多项式的值获得链接补集的 Turaev-Viro 不变量的公式。作为一个应用，我们给出第一个例子来验证陈和第三位作者的体积猜想\,\cite{Chen-Yang}。也就是说，我们证明了图 8 结和 Borromean 环补的 Turaev-Viro 不变量的渐近性决定了相应的双曲体积。我们的计算还展示了彩色琼斯多项式值的渐近行为的新现象，这些现象似乎既不能由 Kashaev-Murakami-Murakami 体积猜想及其各种推广也不能由 Zagier 的量子模块化猜想预测。我们推测任何链接补的 Turaev-Viro 不变量的渐近性决定了链接的单纯体积，并验证了所有具有零单纯体积的结。最后，我们观察到我们的单纯体积猜想在链接的连接和和分裂并集下是稳定的。