Positive torus links T(m,n) and Sym^l-colored torus knots have triply graded Khovanov-Rozansky homology equal to an explicitly defined family of polynomials p(v,w).
Toric braids and $(m,n)$-parking functions
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abstract
The Dyck path algebra construction of Carlsson and Mellit from arXiv:1508.06239 is interpreted as a representation of "the positive part" of the group of toric braids. Then certain sums over $(m,n)$-parking functions are related to evaluations of this representation on some special braids. The compositional $(km,kn)$-shuffle conjecture of Bergeron, Garsia, Leven and Xin from arXiv:1404.4616 is then shown to be a corollary of this relation.
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Torus link homology
Positive torus links T(m,n) and Sym^l-colored torus knots have triply graded Khovanov-Rozansky homology equal to an explicitly defined family of polynomials p(v,w).