Exceptional knot homology (Q2799077)

A quantum invariant associates a Laurent polynomial \(P^{\mathfrak{g},V}(K;q)\in \mathbb{Z}[q^{\pm 1}]\) to a knot \(K\) coloured by a representation \(V\) of \(\mathcal{U}_q(\mathfrak{g})\). A categorification of a quantum invariant is a bi-graded homology theory \(\mathcal{H}^{\mathfrak{g},V}_{i,j}(K; q)\) whose Euler characteristic is \(P^{\mathfrak{g},V}(K;q)\).NEWLINENEWLINEThis paper studies the categorification of the quantum invariant associated to the exceptional Lie algebra \(\mathfrak{e}_6\) and its fundamental \(27\)-dimensional representation for torus knots. This appears to be the first computation of a knot homology associated to an exceptional Lie group.NEWLINENEWLINEFirst, using results of Cherednik, the DAHA-Jones polynomials of some torus knots are computed. These have both positive and negative coefficients, and the structure of differentials on superpolynomials is invoked to fix this. Similar techniques may be applicable to compute torus knot homologies for other exceptional Lie algebras, for supergroups, and for other representations.NEWLINENEWLINEA number of manual computations for \((\mathfrak{e}_6,\mathbf{27})\)-knot homologies appear in this paper; more (computed using the QuaGroup package for GAP) are contained in Appendix D of the first author's dissertation [\textit{R. F. Elliot}, Topological strings, double affine Hecke algebras, and exceptional knot homology. Pasadena, CA: California Institute of Technology (2015), \url{http://thesis.library.caltech.edu/8920/1/Thesis.pdf}].