Dehn fillings of knot manifolds containing essential once-punctured tori

TL;DR

This paper investigates the limits of exceptional Dehn fillings on hyperbolic knot manifolds with essential once-punctured tori, classifying cases with large boundary slope distances and providing a comprehensive understanding of such fillings.

Contribution

It establishes an upper bound of 5 on the distance between slopes leading to Seifert fibered manifolds and classifies all cases where this distance is 4 or more.

Findings

Maximum slope distance for Seifert fibered fillings is 5.

Complete classification of manifolds with slope distance 4 or more.

Identification of unique manifolds corresponding to specific slope pairs.

Abstract

In this paper we study exceptional Dehn fillings on hyperbolic knot manifolds which contain an essential once-punctured torus. Let $M$ be such a knot manifold and let $\beta$ be the boundary slope of such an essential once-punctured torus. We prove that if Dehn filling $M$ with slope $\alpha$ produces a Seifert fibred manifold, then $\Delta(\alpha,\beta)\leq 5$. Furthermore we classify the triples $(M; \alpha,\beta)$ when $\D(\alpha,\beta)\geq 4$. More precisely, when $\D(\alpha,\beta)=5$, then $M$ is the (unique) manifold $Wh(-3/2)$ obtained by Dehn filling one boundary component of the Whitehead link exterior with slope -3/2, and $(\alpha, \beta)$ is the pair of slopes $(-5, 0)$. Further, $\D(\alpha,\beta)=4$ if and only if $(M; \alpha,\beta)$ is the triple $\displaystyle (Wh(\frac{-2n\pm1}{n}); -4, 0)$ for some integer $n$ with $|n|>1$. Combining this with known results, we classify all hyperbolic knot manifolds $M$ and pairs of slopes $(\beta, \gamma)$ on $\partial M$ where $\beta$ is the boundary slope of an essential once-punctured torus in $M$ and $\gamma$ is an exceptional filling slope of distance 4 or more from $\beta$. Refined results in the special case of hyperbolic genus one knot exteriors in $S^3$ are also given.