We demonstrate a variant of the bond fluctuation lattice Monte Carlo model in which moves through cis conformations are forbidden. Ring polymers in this model have a conserved quantity that amounts to a topological linking number. Increased linking number reduces the radius of gyration mildly. A linking number of order 0.2 per bond leads to an 8% reduction of the radius for 128-bond chains. This percentage appears to rise with increasing chain length, contrary to expectation. For ring chains evolving without the conservation of linking number, we demonstrate a substantial anticorrelation between the twist and writhe variables whose sum yields the linking number. We raise the possibility that our observed anticorrelations may have counterparts in the most important practical polymer that conserves linking number, DNA.