University of Sydney Algebra Seminar
Elijah Bodish
Friday 23 August, 12-1pm, Place: Carslaw 275
Spin link homology
Reshetikhin-Turaev define a Laurent polynomial invariant of knots for each simple Lie algebra "colored” by a finite dimensional irreducible representation. In the case of sl(2) and the defining representation, this polynomial invariant is the Jones polynomial.
Khovanov discovered that the Jones polynomial is the Euler characteristic of a complex of graded vector spaces. Thus, Khovanov’s homology categories the Jones polynomial. Many other definitions of categorified Reshetikhin-Turaev invariants have appeared since. The most notable works are: Khovanov-Rozansky’s generalization of Khovanov homology to sl(n), and Webster's uniform construction for an arbitrary simple Lie algebra. However, very little is known (e.g. no examples are computed) unless the Lie algebra is sl(n) and the representation is a fundamental representation.
In my talk I will describe how to equip the sl(2n) link homology, colored by the n-th fundamental representation, with an involution such that the (super) Euler characteristic is the so(2n+1) Reshetikhin-Turaev link polynomial. This construction, which is a priori unrelated to Webster’s, is inspired by folding, categorified skew Howe duality, diagrammatics for centralizer algebras, and iquantum groups.
This is based on arXiv:2407.00189 -- joint work with Ben Elias and David Rose.