\(q,t\)-Catalan numbers and knot homology (Q2918497)

\textit{N. M. Dunfield} et al. [Exp. Math. 15, No. 2, 129--159 (2006; Zbl 1118.57012)] conjectured the existence of a triply graded homology theory that is a categorification of the Homflypt-polynomial, unifying Heegaard-Floer homology and the Khovanov-Rozansky sl(\(N\))-homologies for all \(N\) (in particular Khovanov homology for \(N = 2\), and a trivial homology theory for \(N = 1\)). The main result of this paper is a conjectural algebraic model for this triply graded homology of the (2,3)-, (3,4)- and (4,5)-torus knot. The model is proven to have the correct Euler-characteristic (i.e. the Homflypt-polynomial) and to yield correct results for the trivial homology at \(N = 1\), for Khovanov- and Heegaard-Floer homology.NEWLINENEWLINEAnother conjectural algebraic model is provided for the stabilisation of triply graded homology of the \((n,m)\)-torus knot as \(m\) goes to infinity, specialised at \(q = 1\).NEWLINENEWLINEThe main tools are the deformations of Catalan numbers as polynomials in two variables introduced in [\textit{A. M. Garsia} and \textit{M. Haiman}, J. Algebr. Comb., No. 3, 191--244 (1996; Zbl 0853.05008)], and similar deformations of Schroeder numbers (cf. \textit{J. Haglund} [The \(q,t\)-Catalan numbers and the space of diagonal harmonics. With an appendix on the combinatorics of Macdonald polynomials. University Lecture Series 41. Providence, RI: American Mathematical Society (AMS) (2008; Zbl 1142.05074)]). The relationship to Gukov, Dunfield and Rasmussen's homology is e.g. motivated by the proof that the coefficients of the Homflypt-polynomial of the \((n,m)\)-torus knot are \(q\)-deformations of a number of certain Dyck-paths.NEWLINENEWLINETwo- and three-stranded torus knots and the (4,5)-torus knot are analysed closely.NEWLINENEWLINEFor the entire collection see [Zbl 1242.11004].