  Weight systems on chord diagrams play a central role in knot theory and
Chern-Simons theory; and more recently in stringy quantum gravity. We highlight
that the noncommutative algebra of horizontal chord diagrams is canonically a
star-algebra, and ask which weight systems are positive with respect to this
structure; hence we ask: Which weight systems are quantum states, if horizontal
chord diagrams are quantum observables? We observe that the fundamental
gl(n)-weight systems on horizontal chord diagrams with N strands may be
identified with the Cayley distance kernel at inverse temperature beta=ln(n) on
the symmetric group on N elements. In contrast to related kernels like the
Mallows kernel, the positivity of the Cayley distance kernel had remained open.
We characterize its phases of indefinite, semi-definite and definite
positivity, in dependence of the inverse temperature beta; and we prove that
the Cayley distance kernel is positive (semi-)definite at beta=ln(n) for all
n=1,2,3,... In particular, this proves that all fundamental gl(n)-weight
systems are quantum states, and hence so are all their convex combinations. We
close with briefly recalling how, under our "Hypothesis H", this result impacts
on the identification of bound states of multiple M5-branes.
