Mathematics Colloquia and Seminars

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The colored HOMFLYPT polynomial is a 2-variable invariant of links colored by Young diagrams generalizing the celebrated Jones polynomial and other Type A quantum link polynomials. Constructions of this invariant from a physical perspective reveal a natural symmetry describing its behavior under exchanging each Young diagram coloring a fixed link with its transpose.  

 

One categorical level up, Khovanov and Rozansky constructed a triply-graded homological link invariant that recovers the (trivially colored) HOMFLYPT polynomial upon taking Euler characteristic. Various authors have constructed colored analogues of this invariant and, in special cases, conjectured that their constructions satisfy a categorical lift of this decategorified symmetry. In this talk, we will precisely formulate a version of this conjecture applying to all colorings and outline a recent proof in the special case of a positive torus knot colored by a single row or column of arbitrary length. Time permitting, we will also describe explicit formulas for the column-colored homology of positive torus knots in the framework of singular Soergel bimodules.