The Hogancamp-Mellit recursion, originally used to compute the Khovanov-Rozansky homology of torus knots and links, can also be applied to knots beyond the torus case. In particular, one can compute KR homology of the so-called shortcut torus knots, which also happen to be isotopic to monotone knots of Galashin and Lam corresponding to a straight line with not-necessarily integer end points, or Coxeter knots of triangular partitions in the sense of Oblomkov and Rozansky. Furthermore, the recursion allows one to relate the Poincare polynomials of the KR homology of such knots to triangular Catalan and Schroder polynomials, and to the Shuffle theorem under any line of Blasiak, Haiman, Morse, Pun, and Seelinger.

Most recently, it became apparent that these recursions can be pushed even further, beyond the shortcut torus case. However, the precise extend of applicability and the obstacles to these methods remain unclear.

In this talk, I will introduce the above-mentioned classes of knots, explain how Hogancamp-Mellit recursion applies to them, and how the computation is related to the Shuffle theorems. I will also demonstrate computations for Khovanov-Rozansky homology beyond the shortcut torus case. The talk is based on a joint paper with Carmen Caprau, Nicolle Gonzales, and Matt Hogancamp. The computational tools for knots beyond the shortcut torus knots were developed by Andrei Mazin as a part of a summer undergraduate research project.