Pseudo-Anosov maps and surgery on fibred 2-bridge knots (Q750951)

The authors use laminations to study fibred knots in the 3-sphere, mainly focusing on fibred 2-bridge knots. For these, the monodromy is a product of Dehn twists about loops in the fiber which are invariant under an involution whose quotient is the 2-disc. This is used to give an explicit construction of an invariant train track for the monodromy, and thereby a lamination invariant under a homeomorphism g isotopic to the monodromy. When g is pseudo-Anosov (which is the case whenever the knot is not a torus knot) Gabai has used it to define an invariant of the knot, called the degeneracy type. It measures the amount by which g rotates the boundary component of the fiber. A theorem of Gabai and Oertel gives conditions, in terms of this degeneracy, which guarantee that (a,b)- surgery on the knot will yield a manifold whose universal cover if \({\mathbb{R}}^ 3\). The authors' analysis of the 2-bridge case calculates the degeneracy type, in particular showing that all nontrivial surgeries on nontorus fibred 2-bridge knots yield manifolds with universal cover \({\mathbb{R}}^ 3\).