The abelian group of $k$chord diagrams on a skeleton of two directed line segments (modulo the STU relation), $\mathcal A_k(\uparrow\uparrow)$, is known to have $2$torsion when $k=5$. In fact, I know that in his 1998 thesis "Eine Abhandlung über die Algebra der Schlingeldiagramme," Ilya Dogolazky showed that $\mathcal A_5(\uparrow\uparrow)\cong\mathbb Z^{148}\oplus \mathbb Z_2$. Ted Stanford showed that this $\mathbb Z_2$ cannot be promoted to an invariant of $2$string links, so that there is no "mod 2 Kontsevich integral." My question is whether anyone has calculated $\mathcal A_6(\uparrow\uparrow),$ whether there's any torsion in it, and whether it is known if that torsion lifts to invariants of $2$string links?
