Row-Column Mirror Symmetry for Colored Torus Knot Homology

Abstract

We give a recursive construction of the categorified Young symmetrizer introduced by Abel-Hogancamp in arXiv:1510.05330 corresponding to the single-column partition. As a consequence, we obtain new expressions for the uncolored $y$-ified HOMFLYPT homology of positive torus links and the $y$-ified column-colored HOMFLYPT homology of positive torus knots. In the latter case, we compare with the row-colored homology of positive torus knots computed by Hogancamp-Mellit in arXiv:1909.00418, verifying the mirror symmetry conjectures of arXiv:1112.0030 and arXiv:1304.3481 in this case.

87 pages; many figures. v2: Final version to appear in Selecta Mathematica. Section 5 significantly expanded and all material on singular Soergel bimodules moved to a new Appendix. Many revisions throughout to improve clarity and correct typos