The number of convex polygons on the square and honeycomb lattices

Abstract

A subset of the set of self-avoiding polygons (SAP) embeddable on the square lattice which display the property of convexity is defined. An algorithm for their enumeration is developed, and from the available series coefficients the exact generating function is found. The singularity structure appears similar to that of the unsolved SAP problem, but with different critical exponents and critical points. The enumeration of convex polygons allows the extension of the existing series for the square lattice SAP by one term. For the honeycomb lattice similar results have been obtained, despite a less natural definition of convexity.