Immmersions of the circle in the sphere and in higher Riemann surfaces
Speaker: Robert Coquereaux, Centre de Physique ThÃ©orique (CPT)
Abstract: We encode circle immersions with n crossings in terms of orbits of the centraliser of a special element of the symmetric groups S(2n) or S(4n) acting by conjugation on particular subsets, or in terms of appropriate double cosets. The details depend on the various orientability hypothesis made on the source (the circle) and on the target (a surface of genus g), and also on a possible constraint of bi-colariability that one can furthermore impose. We count and tabulate non-equivalent images of spherical immersions up to 10 crossings therefore recovering and extending results by Arnold (5 crossings) and followers (7 crossings), we also obtain the corresponding classifications for genus higher than 0. In the latter case we introduce the notion of bicolourability and determine the first terms (up to 9 crossings) of the corresponding sequences. This presentation summarizes recent work done in collaboration with with J.-B. Zuber.
Room Reservation Information
Room Number: 106 McAllister
Time: 2:30pm - 3:30pm