Guts, Dehn Fillings and Volumes of Hyperbolic Manifolds
We construct an invariant called guts for second homology classes in irreducible 3-manifolds with toral boundary and non-degenerate Thurston norm. We prove that guts of second homology classes in each Thurston cone are invariant under a natural condition. We show that the guts of different homology classes are related by sutured decompositions. As an application, an invariant of knot complements is given and is computed in a few interesting cases.
The minimal volume of orientable hyperbolic manifold with a given number of cusps has been found for 0,1,2,4 cusps, while the minimal volume of 3-cusped orientable hyperbolic manifolds remains unknown. By using guts in sutured manifolds and pared manifolds, we are able to show that for an orientable hyperbolic 3-manifold with 3 cusps such that every second homology class is libroid, its volume is at least 5.49... = 6 times Catalan’s constant.
In the final chapter, we develop a method for controlling the upper bound of the Euler characteristic of surfaces in sufficiently long Dehn fillings. By using this method, we show that for a compact orientable irreducible acoannular 3-manifold with toral boundary, properly norm-minimizing surfaces capped off with disks in sufficiently long fillings are still properly norm-minimizing. We also show that if a sufficiently long Dehn surgery on a link in a product sutured manifold yields the same product sutured manifold, then this link is horizontal in the product sutured manifold. Combining these results, we prove that for a compact orientable irreducible acoannular 3-manifold with toral boundary, the cores of its sufficiently long fibered Dehn fillings are either horizontal or vertical. Moreover, we provide a new way to prove that there are finitely many links with a given complement such that no component is unknotted and no 2-component sublink is coaxial. Another result coming from this method is that all sufficiently long fillings of an orientable, irreducible, boundary-irreducible-irreducible, acoannular 3-manifold with toral boundary are still irreducible, boundary-irreducible-irreducible, acoannular.