In this paper we propose and discuss implications of a general conjecture that there is a natural action of a rank 1 double affine Hecke algebra on the Kauffman bracket skein module of the complement of a knot $K\subset S^{3}$. We prove this in a number of nontrivial cases, including all $(2,2p+1)$ torus knots, the figure eight knot, and all 2-bridge knots (when $q=\pm 1$). As the main application of the conjecture, we construct three-variable polynomial knot invariants that specialize to the classical colored Jones polynomials introduced by Reshetikhin and Turaev. We also deduce some new properties of the classical Jones polynomials and prove that these hold for all knots (independently of the conjecture). We furthermore conjecture that the skein module of the unknot is a submodule of the skein module of an arbitrary knot. We confirm this for the same example knots, and we show that this implies that the colored Jones polynomials of $K$ satisfy an inhomogeneous recursion relation.

Khovanov homology, an invariant of links in ${ \mathbb{R} }^{3} $, is a graded homology theory that categorifies the Jones polynomial in the sense that the graded Euler characteristic of the homology is the Jones polynomial. Asaeda et al. [‘Categorification of the Kauffman bracket skein module of $I$-bundles over surfaces’, Algebr. Geom. Topol. 4 (2004), 1177–1210] generalised this construction by defining a double graded homology theory that categorifies the Kauffman bracket skein module of links in $I$-bundles over surfaces, except for the surface $ \mathbb{R} {\mathrm{P} }^{2} $, where the construction fails due to strange behaviour of links when projected to the nonorientable surface $ \mathbb{R} {\mathrm{P} }^{2} $. This paper categorifies the missing case of the twisted $I$-bundle over $ \mathbb{R} {\mathrm{P} }^{2} $, $ \mathbb{R} {\mathrm{P} }^{2} \widetilde {\times } I\approx \mathbb{R} {\mathrm{P} }^{3} \setminus \{ \ast \} $, by redefining the differential in the Khovanov chain complex in a suitable manner.

In this paper we present a short definition of the Witten invariants of 3-manifolds. We also give simple proofs of invariance of those obtained for r = 3 and r = 4. Our definition is extracted from the 1993 paper of Lickorish and the Prasolov-Sossinsky book, where it is dispersed over 20 pages. We show by several examples that it is indeed convenient for calculations.