Partial duality of hypermaps

Sergei Chmutov, Fabien Vignes-Tourneret

Published 2014-09-02Version 1

We introduce a collection of new operations on hypermaps, partial duality, which include the classical Euler-Poincar\'e dualities as particular cases. These operations generalize the partial duality for maps, or ribbon graphs, recently discovered in a connection with knot theory. Partial duality is different from previous studied operations of S. Wilson, G. Jones, L. James, and A. Vince. Combinatorially hypermaps may be described in one of three ways: as three involutions on the set of flags ($\tau$-model), or as three permutations on the set of half-edges ($\sigma$-model in orientable case), or as edge 3-colored graphs. We express partial duality in each of these models.