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?
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$\begingroup$ Hi Jim, the string link monoid does not have any torsion. Or are you talking about homotopy string links? $\endgroup$– Ryan BudneyMay 19, 2014 at 21:09
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$\begingroup$ @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. $\endgroup$– Jim ConantMay 20, 2014 at 1:56
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