Sum rules for the ground states of the O(1) loop model on a cylinder and   the XXZ spin chain
  The sums of components of the ground states of the O(1) loop model on a cylinder or of the XXZ quantum spin chain at Delta=-1/2 (of size L) are expressed in terms of combinatorial numbers. The methods include the introduction of spectral parameters and the use of integrability, a mapping from size L to L+1, and knot-theoretic skein relations. 
P. Di Francesco, P. Zinn-Justin and J.-B. Zuber