Degeneracy slopes, boundary slopes and exceptional surgery slopes

TL;DR

This paper establishes bounds relating degeneracy slopes, boundary slopes, and exceptional surgery slopes in 3-manifolds, with applications to knot theory and confirmation of conjectures for hyperbolic knots.

Contribution

It provides new bounds on slopes in 3-manifolds, confirming parts of conjectures and extending previous results on Montesinos knots and hyperbolic knots.

Findings

Degeneracy slope for a full essential lamination in knot exterior is meridional.

Bounds on boundary slopes' denominators and differences for hyperbolic knots.

Bounds on exceptional surgery slopes' denominators and range, supporting conjectures.

Abstract

We give an upper bound on the distance between a degeneracy slope for a very full essential lamination and a boundary slope of an essential surface embedded in a compact, orientable, irreducible, atoroidal 3-manifold with incompressible torus boundary. There are three applications: (i) We show that a degeneracy slope for a very full essential lamination in the exterior of a prime alternating knot is meridional. This gives an affirmative answer to part of a conjecture posed by Gabai and Kazez. (ii) We obtain two bounds on boundary slopes for a hyperbolic knot in an integral homology sphere, at least one of which always holds: one concerning the denominators of boundary slopes, and the other concerning the differences between boundary slopes. This generalizes a result on Montesinos knots obtained by the author and Mizushima. (iii) We obtain two bounds on exceptional surgery slopes for a hyperbolic knot in an integral homology sphere, at least one of which always holds: one concerning the denominators of such slopes, and the other concerning their range in terms of the genera of the knots. Both are actually conjectured by Gordon and Teragaito to always hold for hyperbolic knots in the 3-sphere.