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Geometric Methods in Rep Theory Seminar – Luke Conners (UNC)
February 9 @ 4:00 pm - 5:00 pm
Row-Column Mirror Symmetry for Colored Torus Knot Homology
Abstract. The HOMFLYPT polynomial is a 2-variable link invariant generalizing the celebrated Jones polynomial and other Type A quantum link polynomials. Its construction passes through a Hecke algebra representation of the braid group, and by making use of certain idempotent elements in the Hecke algebra, one can extend the invariant to links with components labeled by arbitrary Young diagrams. The resulting invariant, called the colored HOMFLYPT polynomial, has a well-known “mirror symmetry” property describing its behavior under exchanging each such Young diagram with its transpose. One categorical level up, Khovanov and Rozansky constructed a triply-graded homological link invariant that recovers the HOMFLYPT polynomial upon taking Euler characteristic. Various authors have constructed colored versions of triply-graded Khovanov-Rozansky homology, and these invariants are conjectured to satisfy a categorical lift of the polynomial mirror symmetry described above. In this talk, we will formulate this conjecture precisely and outline a recent proof in the special case of a positive torus knot colored by a single row or column of arbitrary length.