Knot probability for lattice polygons in confined geometries

We study the knot probability of polygons confined to slabs or prisms, considered as subsets of the simple cubic lattice. We show rigorously that almost all sufficiently long polygons in a slab are knotted and we use Monte Carlo methods to investigate the behaviour of the knot probability as a function of the width of the slab or prism and the number of edges in the polygon. In addition we consider the effect of solvent quality on the knot probability in these confined geometries.