Maps, immersions and permutations

We consider the problem of counting and of listing topologically inequivalent "planar" {4-valent} maps with a single component and a given number n of vertices. This enables us to count and to tabulate immersions of a circle in a sphere (spherical curves), extending results by Arnold and followers. Different options where the circle and/or the sphere are/is oriented are considered in turn, following Arnold's classification of the different types of symmetries. We also consider the case of bicolourable and bicoloured maps or immersions, where faces are bicoloured. Our method extends to immersions of a circle in a higher genus Riemann surface. There the bicolourability is no longer automatic and has to be assumed. We thus have two separate countings in non zero genus, that of bicolourable maps and that of general maps. We use a classical method of encoding maps in terms of permutations, on which the constraints of "one-componentness" and of a given genus may be applied. Depending on the orientation issue and on the bicolourability assumption, permutations for a map with n vertices live in S(4n) or in S(2n). In a nutshell, our method reduces to the counting (or listing) of orbits of certain subset of S(4n) (resp. S(2n)) under the action of the centralizer of a certain element of S(4n) (resp. S(2n)). This is achieved either by appealing to a formula by Frobenius or by a direct enumeration of these orbits. Applications to knot theory are briefly mentioned.Comment: 46 pages, 18 figures, 9 tables. Version 2: added precisions on the notion used for the equivalence of immersed curves, new references. Version 3: Corrected typos, one array in Appendix B1 was duplicated by mistake, the position of tables and the order of the final sections have been modified, results unchanged. To be published in the Journal of Knot Theory and Its Ramification

Plethysm and lattice point counting

We apply lattice point counting methods to compute the multiplicities in the plethysm of $GL(n)$. Our approach gives insight into the asymptotic growth of the plethysm and makes the problem amenable to computer algebra. We prove an old conjecture of Howe on the leading term of plethysm. For any partition $\mu$ of 3,4, or 5 we obtain an explicit formula in $\lambda$ and $k$ for the multiplicity of $S^\lambda$ in $S^\mu(S^k)$.Comment: 25 pages including appendix, 1 figure, computational results and code available at http://thomas-kahle.de/plethysm.html, v2: various improvements, v3: final version appeared in JFoC

Counting Orbifolds

We present several methods of counting the orbifolds C^D/Gamma. A correspondence between counting orbifold actions on C^D, brane tilings, and toric diagrams in D-1 dimensions is drawn. Barycentric coordinates and scaling mechanisms are introduced to characterize lattice simplices as toric diagrams. We count orbifolds of C^3, C^4, C^5, C^6 and C^7. Some remarks are made on closed form formulas for the partition function that counts distinct orbifold actions.Comment: 69 pages, 9 figures, 24 tables; minor correction