The physical 3d $\mathcal N = 2$ theory $T[Y]$ was previously used to predict the existence of some $3$-manifold invariants $\widehat{Z}_{a}(q)$ that take the form of power series with integer coefficients, converging in the unit disk. Their radial limits at the roots of unity should recover the Witten–Reshetikhin–Turaev invariants. In this paper we discuss how, for complements of knots in $S^3$, the analogue of the invariants $\widehat{Z}_{a}(q)$ should be a two-variable series $F_K(x,q)$ obtained by parametric resurgence from the asymptotic expansion of the colored Jones polynomial. The terms in this series should satisfy a recurrence given by the quantum A-polynomial. Furthermore, there is a formula that relates $F_K(x,q)$ to the invariants $\widehat{Z}_{a}(q)$ for Dehn surgeries on the knot. We provide explicit calculations of $F_K(x,q)$ in the case of knots given by negative definite plumbings with an unframed vertex, such as torus knots. We also find numerically the first terms in the series for the figure-eight knot, up to any desired order, and use this to understand $\widehat{Z}_a(q)$ for some hyperbolic 3-manifolds.

中文翻译：

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结补码的两变量级数

以前使用物理3d $ \ math N = 2 $理论$ T [Y] $来预测某些$ 3 $流形不变量$ \ widehat {Z} _ {a}（q）$的存在形式具有整数系数的幂级数，收敛在单位圆盘中。它们在统一根源处的径向极限应能恢复维滕-雷什蒂欣-图拉耶夫不变式。在本文中，我们讨论了对于$ S ^ 3 $中的结的补码，不变量$ \ widehat {Z} _ {a}（q）$的类似物应如何是二变量系列$ F_K（x，q ）$是通过有色的琼斯多项式的渐近展开式通过参数回潮获得的。该系列中的术语应满足量子A多项式给出的递归。此外，有一个公式将$ F_K（x，q）$与结上Dehn手术的不变量$ \ widehat {Z} _ {a}（q）$相关。我们提供$ F_K（x，q）$，如果结点是由带有无框顶点的负定水暖给定的，例如圆环结。我们还从数字上找到了八字形结系列中的第一项，达到任意所需的阶数，并使用它来理解某些双曲型3流形的$ \ widehat {Z} _a（q）$。