Degeneracy slopes, boundary slopes and exceptional surgery slopes

TL;DR

The paper investigates how close a degeneracy slope $\Delta(\beta, \delta)$ of a very full essential lamination can be to the boundary slope of an essential surface in knot exteriors, establishing the bound $\Delta(\beta, \delta) \le 2 \frac{-\chi(F)}{|\partial F|}$ (and $\le 4 g(F) - 2$ when $F$ is orientable). The main method uses laminations carried by essential branched surfaces, splitting to obtain essential train tracks, and a cusped Euler characteristic argument to derive the bound. This leads to three applications: (i) degeneracy slopes for alternating knots are meridional, (ii) two slope-bounds for boundary slopes of hyperbolic knots in integral homology spheres, and (iii) two bounds on exceptional surgery slopes; together these extend known results and address conjectures in the field. The results connect degeneracy phenomena with Dehn surgery behavior, providing new constraints on boundary slopes and exceptional surgeries in hyperbolic knot complements.

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.

Theorems & Definitions (17)

• Theorem 1
• Theorem 2: Degeneracy slopes for alternating knots
• Corollary 1
• proof
• Theorem 3: Bounds on boundary slopes
• Theorem 4: Bounds on exceptional surgery slopes
• Corollary 2
• proof
• Lemma 1: Wu
• proof : Proof of Theorem \ref{['Thm1']}