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Dynamics on 3-manifolds and twisted Alexander polynomials


Speaker: Anna Parlak, UC Davis
Location: 2112 MSB
Start time: Thu, Nov 3 2022, 2:10PM

For certain pseudo-Anosov flows one can define a polynomial invariant which informs about the dynamics of the flow. A classical example is the Teichmüller polynomial associated to a fibered face of the Thurston norm ball. Its main feature is that it packages information about the stretch factors of monodromies of all fibrations lying over that face.

Until recently the Teichmüller polynomial was rather tricky to compute. All known algorithms were either hard to implement or limited to some special families of fibered 3-manifolds. I will discuss a theorem which says that the Teichmüller polynomial is always a specialization of a certain twisted Alexander polynomial of some surgery parent of the original manifold. This result implies various algebraic properties of the Teichmüller polynomials, and gives a new quick algorithm to compute the Teichmüller polynomial using Fox calculus.