Squeezing knots

A model of knotted polymers in a confined space is studied by considering lattice polygons of fixed knot type in a slab geometry. If p(n)(K,w) is the number of lattice polygons of knot type K in a slab of width w, then the generating function of this lattice model is given by g(K)(w;t) = Sigma(infinity)(n=0) p(n)(K, w)t(n), where t is a generating variable conjugate with the length of the polygon. In this paper the scaling of the mean length < n >(K, w) of polygons in this ensemble is examined. A Metropolis Monte Carlo implementation of the BFACF algorithm is shown to be ergodic in this ensemble, and it is used to estimate the mean lengths of knotted polygons in slabs of various widths. Our numerical results are consistent with the predictions of the scaling arguments. In addition, the metric properties of knotted polygons in this ensemble are examined numerically.