
## 2. Parity

1. #### Problem 2.1.

Call a braid $\beta$ parity if HHH$(\beta)$ is parity. Find necessary and/or sufficient conditions for a braid to be parity.

Are there operations on braids (for example, adding a positive number of full twists) which transform it into a parity braid or which preserve parity braids?
• #### Problem 2.2.

Are algebraic braids parity? If so find a (recursive) formula for HHH of an algebraic braid.
• #### Problem 2.3.

There is a commutative family of braids built as products of FT$_k$, the full twist on the first $k$ strands, for $k \leq n$. Are the positive products in this family parity? If so, find a (recursive) formula for HHH.
• #### Problem 2.4.

In the presence of parity, there are two combinatorial formulas for HHH: one for localization of Hilbert schemes from the GNR conjecture and one from the Elias-Hogencamp-Mellit papers. How are these formulas related? How are they related to Demazure crystals?

Cite this as: AimPL: Categorified Hecke algebras, link homology, and Hilbert schemes, available at http://aimpl.org/catheckehilbert.