Enumeration of r-regular maps on the torus. Part I: Rooted maps on the torus, the projective plane and the Klein bottle. Sensed maps on the torus

Enumeration of unsensed orientable and non-orientable maps

“…His approach reduces the enumerating problem for sensed maps on the sphere to counting quotient maps on orbifolds, maps on quotients of a surface under a finite group of automorphisms. His ideas were further developed in a series of papers devoted to enumeration of sensed maps on orientable surfaces of a given genus g [3], regular sensed maps on the torus [4], regular sensed maps on orientable surfaces of a given genus g [5], sensed hypermaps [6], one-face regular sensed maps [7] and one-face maximal unsensed maps [8]. In all these papers a geometric approach based on enumeration of rooted quotient maps on cyclic orbifolds was employed.…”

“…For the example of the orbifold O shown in Figure 1, the branch points a and c have branch indices equal to 4 and the branch index of b is equal to 2. Consequently, the signature of the corresponding orbifold takes the form O(0; [2,4,4]) ≡ O(0; [2, 4 2 ]).…”

Enumeration of 3-regular one-face maps on orientable or non-orientable surface up to all symmetries

“…It is easy to see that in this case the Proposition 3.3 still holds true. Together with the Proposition 4.1 and the Riemann-Hurwitz formula (7) this fact allows to derive the following statement.…”

“…Their approach reduces the enumerating problem for sensed maps on a surface to counting quotient maps on orbifolds, rooted maps on quotients of this surface under a finite group of automorphisms. Their ideas were further developed in a series of papers devoted to enumeration of sensed hypermaps [4], one-face regular sensed maps [5], one-face maximal unsensed maps [6], regular sensed maps on the torus [7] and regular sensed maps on orientable surfaces of a given genus g [8].…”

The Fourth Fundamental Form of the Torus Hypersurface

“…Surfaces and hypersurfaces have been worked by the mathematicians for centuries. We see some new papers about torus surfaces and torus hypersurfaces in the literature such as [2][3][4][5][6][7][8][9][10][11][12][13][14][15].…”