An Introduction to the Volume Conjecture and Its Generalizations

In this paper we give an introduction to the volume conjecture and its generalizations. Especially we discuss relations of the asymptotic behaviors of the colored Jones polynomials of a knot with different parameters to representations of the fundamental group of the knot complement at the special linear group over complex numbers by taking the figure-eight knot and torus knots as examples. After V. Jones’ discovery of his celebrated polynomial invariant V (K; t) in 1985 [22], Quantum Topology has been attracting many researchers; not only mathematicians but physicists. The Jones polynomial was generalized to two kinds of two-variable polynomials, the HOMFLYpt polynomial [10, 52] and the Kauffman polynomial [27] (see also [21, 3] for another one-variable specialization). It turned out that these polynomial invariants are related to quantum groups introduced by V. Drinfel′d and M. Jimbo (see for example [26, 55]) and their representations. For example the Jones polynomial comes from the quantum group Uq(sl2(C)) and its two-dimensional representation. We can also define the quantum invariant associated with a quantum group and its representation. If we replace the quantum parameter q of a quantum invariant (t in V (K; t)) with e we obtain a formal power series in the formal parameter h. Fixing a degree d of the parameter h, all the degree d coefficients of quantum invariants share a finiteness property. By using this property, one can define a notion of finite type invariant [2, 1]. (See [57] for V. Vassiliev’s original idea.) Via the Kontsevich integral [35] (see also [36]) we can recover a quantum invariant from the corresponding ‘classical’ data. (Note that a quantum group is a deformation of a ‘classical’ Lie algebra.) So for example one only needs to know the Lie algebra sl2(C) (easy!) and its fundamental two-dimensional representation (very easy!) to define the Jones polynomial, provided that one knows the Kontsevich integral (unfortunately, this is very difficult). In the end of the 20th century, M. Khovanov introduced yet another insight to Quantum Topology. He categorified the Jones polynomial and defined a homology for a knot such that its graded Euler characteristic coincides with the Jones polynomial [29]. See [30] for a generalization to the HOMFLYpt polynomial. Now we are in the 21st century. In 2001, J. Murakami and the author proposed the Volume Conjecture [48] to relate a series of quantum invariants, the N -colored Jones polynomials of a knot, to the volume of the knot complement, generalizing R. Kashaev’s conjecture [24]. The aim of this article is to introduce the conjecture and some of its generalizations, emphasizing a relation of the asymptotic behavior of the series of the colored Jones Date: February 4, 2008. 2000 Mathematics Subject Classification. Primary 57M27, Secondary 57M25 57M50 58J28.