Entropic exponents of lattice polygons with specified knot type

Ring polymers in three dimensions can be knotted, and the dependence of their critical behaviour on knot type is an open question. We study this problem for polygons on the simple cubic lattice using a novel grand-canonical Monte Carlo method and present numerical evidence that the entropic exponent depends on the knot type of the polygon. We conjecture that the exponent increases by unity for each additional factor in the knot factorization of the polygon.