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

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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.