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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 ├╝ber 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?

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Hi Jim, the string link monoid does not have any torsion. Or are you talking about homotopy string links? –  Ryan Budney May 19 at 21:09
    
@RyanBudney: whether the string link monoid has torsion is not relevant. The question is about the space of chord diagrams, and whether the torsion there "integrates" to a torsion-valued invariant. As I mentioned in the comment, it is known that there is 2-torsion in the abelian group of 5 chord diagrams, but it doesn't correspond to an invariant. This implies that the 4T and 1T relations are not sufficient for string links, even though conjecturally they are for knots. –  Jim Conant May 20 at 1:56
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