Dehn surgery on knots in $S^3$ producing Nil Seifert fibred spaces

TL;DR

This paper classifies all Nil Seifert fibred spaces obtainable via Dehn surgery on non-trefoil knots in $S^3$, identifying six specific spaces and proposing a conjecture about hyperbolic knots with such surgeries.

Contribution

It precisely determines the six Nil Seifert fibred spaces from non-trefoil knots and introduces a conjecture on four hyperbolic knots admitting Nil Seifert fibred surgeries.

Findings

Exactly six Nil Seifert fibred spaces from non-trefoil knots in $S^3$

The set of all such surgery slopes is $oxed{60, 144, 156, 288, 300}$ and their mirrors

Conjecture: four hyperbolic knots in $S^3$ admit Nil Seifert fibred surgeries

Abstract

We prove that there are exactly $6$ Nil Seifert fibred spaces which can be obtained by Dehn surgeries on non-trefoil knots in $S^3$, with $\{60, 144, 156, 288, 300\}$ as the exact set of all such surgery slopes up to taking the mirror images of the knots. We conjecture that there are exactly $4$ specific hyperbolic knots in $S^3$ which admit Nil Seifert fibred surgery. We also give some more general results and a more general conjecture concerning Seifert fibred surgeries on hyperbolic knots in $S^3$.