On spun-normal and twisted squares surfaces

Yoshida in [6] and Tillmann in [3] describe di erent methods of producing surfaces within a 3-manifoldM with boundary from ideal points of the deformation variety of M , given a particular ideal tetrahedralisation of M . The Yoshida construction builds a surface from twisted squares within the tetrahedra, the Tillmann construction results in a spun-normal surface relative to the tetrahedra. In this paper we investigate the connection between the methods. We show rst that a twisted squares surface can be obtained from the spun-normal surface by an isotopy and possibly, two di erent types of compression moves (the usual disk compression moves, and compressions along certain annuli).