The realization problem for non-integer Seifert fibered surgeries

TL;DR

This paper advances understanding of which knots in the 3-sphere can produce small Seifert fibered spaces via non-integer surgeries, showing that under certain conditions, such spaces come from torus knots or their cables.

Contribution

It proves that for specific Seifert fibered spaces with certain plumbing weights, the spaces can be realized through non-integer surgeries on torus knots or their cables.

Findings

Spaces with certain plumbing weights are realizable via torus knot surgeries.

Progress on the conjecture about non-integer surgeries producing Seifert fibered spaces.

Identification of conditions under which Seifert fibered spaces arise from specific knot surgeries.

Abstract

Conjecturally, the only knots in $S^3$ with non-integer surgeries producing Seifert fibered spaces are torus knots and cables of torus knots. In this paper, we make progress on the associated realization problem. Let $Y$ be a small Seifert fibered space arising by $p/q$-surgery on a knot in $S^3$, where $p/q$ is positive and a non-integer. Let $e$ denote the weight of the central vertex in the minimal star-shaped plumbing that $Y$ bounds. We show that if $e\leq -2$ or $e\geq 3$, then $Y$ can be obtained by $p/q$-surgery on a torus knot or a cable of a torus knot.