Jaeger's Higman-Sims state model and the B_2 spider
  Jaeger [Geom. Dedicata 44 (1992), 23-52] discovered a remarkable checkerboard state model based on the Higman-Sims graph that yields a value of the Kauffman polynomial, which is a quantum invariant of links. We present a simple argument that the state model has the desired properties using the combinatorial $B_2$ spider [Comm. Math. Phys. 180 (1996), 109-151]. 
Greg Kuperberg (UC Davis)