Algebra Seminar: Elijah Bodish -- Spin link homology

Elijah Bodish (MIT) will be speaking in the algebra seminar this week.  We will go out
to lunch after the talk.  

Where: Carslaw 275 

When: 12-1pm, Friday 23 August 

Title: Spin link homology 

Abstract: 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.