Characteristic submanifold theory and toroidal Dehn filling

TL;DR

This paper proves a bound on the distance between certain types of Dehn fillings on hyperbolic knot manifolds, using topological and combinatorial analysis of embedded surfaces.

Contribution

It verifies the exceptional Dehn filling conjecture for cases involving small Seifert and toroidal slopes under specific topological conditions.

Findings

Bound on the distance between slopes: Δ(α, β) ≤ 5

Analysis of intersection graphs of immersed surfaces in Dehn fillings

Use of characteristic subsurfaces to understand manifold topology

Abstract

The exceptional Dehn filling conjecture of the second author concerning the relationship between exceptional slopes $\alpha, \beta$ on the boundary of a hyperbolic knot manifold $M$ has been verified in all cases other than small Seifert filling slopes. In this paper we verify it when $\alpha$ is a small Seifert filling slope and $\beta$ is a toroidal filling slope in the generic case where $M$ admits no punctured-torus fibre or semi-fibre, and there is no incompressible torus in $M(\beta)$ which intersects $\partial M$ in one or two components. Under these hypotheses we show that $\Delta(\alpha, \beta) \leq 5$. Our proof is based on an analysis of the relationship between the topology of $M$, the combinatorics of the intersection graph of an immersed disk or torus in $M(\alpha)$, and the two sequences of characteristic subsurfaces associated to an essential punctured torus properly embedded in $M$.