Quantum Knot Invariant for Torus Link and Modular Forms

Recent studies reveal an intimate connection between the quantum knot invariant and the “nearly modular form” especially with the half integral weight. In Ref. 8 Lawrence and Zagier studied an asymptotic expansion of the Witten–Reshetikhin–Turaev invariant of the Poincaré homology sphere, and they showed that the invariant can be regarded as the Eichler integral of the modular form of weight 3/2. In Ref. 18, Zagier further studied a “strange identity” related to the halfderivatives of the Dedekind η-function, and clarified a role of the Eichler integral with half-integral weight. From the viewpoint of the quantum invariant, Zagier’s q-series was originally connected with a generating function of an upper bound of the number of linearly independent Vassiliev invariant [16], and later it was found that Zagier’s q-series with q being the N-th root of unity coincides with Kashaev’s invariant [4,5], which was shown [13] to coincide with a specific value of the colored Jones function, for the trefoil knot. This correspondence was further investigated for the torus knot, and it was shown [2] that Kashaev’s invariant for the torus knot T (2, 2 m+ 1) also has a nearly modular property; it can be regarded as a limit q being the root of unity of the Eichler integral of the Andrews–Gordon q-series, which is theta series with weight 1/2 spanning m-dimensional space. As the torus knot is not hyperbolic, studies of the torus knot may not be attractive from the “Volume Conjecture” [4,13] which states that an asymptotic limit of Kashaev’s invariant coincides with the hyperbolic volume of the knot complement, but they are rather absorbing from the point of view of the number theory, q-series, and modular form.