Normal surfaces and Thurston's equation on triangulated 3-manifolds

Feng Luo, Rutgers New Brunswick

We show that for any closed orientable triangulated pseudo 3-manifold, there exists either a solution to generalized Thurston's gluing equation or a solution to Haken's normal surface equation so that exactly one or two of its quadrilateral coordinates are none zero (a 2-quad-type solution). The proof uses circle-valued angle structures and the volume optimization. If time permits, we will also report our recent joint work with Stephan Tillmann on the topology of minimally triangulated closed 3-manifolds which support 2-quad-type solutions to Haken's equation.