We investigate properties of a multivariate function E(m1,m2, . . . ,mr), called orbicyclic, that arises in enumerative combinatorics in counting non-isomorphic maps on orientable surfaces. E(m1,m2, . . . ,mr) proves to be multiplicative, and a simple formula for its calculation is provided. It is shown that the necessary and sufficient conditions for this function to vanish are equivalent to f...

Let X be a closed oriented Riemann surface of genus ≥ 2 of constant negative curvature −1. A surface containing a disk of maximal radius is an optimal surface. This paper gives exact formulae for the number of optimal surfaces of genus ≥ 4 up to orientation-preserving isometry. We show that the automorphism group of such a surface is always cyclic of order 1, 2, 3 or 6. We also describe a combi...

We consider irreducible 3–manifolds M that arise as knot complements in closed 3–manifolds and that contain at most two connected strict essential surfaces. The results in the paper relate the boundary slopes of the two surfaces to their genera and numbers of boundary components. Explicit quantitative relationships, with interesting asymptotic properties, are obtained in the case that M is a kn...