Self-avoiding surfaces with knotted boundaries

Abstract

The authors consider orientable self-avoiding surfaces, with genus g, embedded in the hypercubic lattice. They consider a surface which has a boundary component and non-zero genus, and devise a construction which will reduce the genus of the surface. This result enables them to study embeddings of surfaces in the three-dimensional lattice, where a surface of genus g may have a boundary component which is a knot of genus g'<or=g. They prove that the growth constants of these surfaces are independent of the knot type of the boundary component, and they derive inequalities between the associated critical exponents.