2. Parity

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 EliasHogencampMellit 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.