Incompressible surfaces in hyperbolic punctured torus bundles are strongly detected

Culler and Shalen, and later Yoshida, give ways to construct incompressible surfaces in 3-manifolds from ideal points of the character and tetrahedron variety, respectively. We work in the case of hyperbolic punctured torus bundles, for which the incompressible surfaces were classified by Floyd and Hatcher. We convert incompressible surfaces from their form to the form output by Yoshida's construction, and effectively run his construction backwards, which gives us the data needed to construct ideal points of the tetrahedron variety corresponding to those surfaces. We note that the constructed ideal points of the tetrahedron variety give us corresponding ideal points of the character variety.;Certain incompressible surfaces, namely the fiber and semi-fibers, cannot be dealt with in this way. We identify which incompressible surfaces are semi-fibers and construct ideal points for them and the fiber for the Culler-Shalen version of the construction.;The existence of these ideal points shows that all incompressible surfaces in hyperbolic punctured torus bundles are strongly detected by the character variety. In other words, all such surfaces are constructed by the Culler-Shalen method.