Finite Dehn surgeries on knots in $S^3$

Yi Ni; Xingru Zhang

doi:10.2140/agt.2018.18.441

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arXiv · 1607.05685

Finite Dehn surgeries on knots in $S^3$

Yi Ni·Xingru Zhang

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Abstract

We show that on a hyperbolic knot $K$ in $S^3$, the distance between any two finite surgery slopes is at most two and consequently there are at most three nontrivial finite surgeries. Moreover in case that $K$ admits three nontrivial finite surgeries, $K$ must be the pretzel knot $P(-2,3,7)$. In case that $K$ admits two noncyclic finite surgeries or two finite surgeries at distance two, the two surgery slopes must be one of ten or seventeen specific pairs respectively. For $D$-type finite surgeries, we improve a finiteness theorem due to Doig by giving an explicit bound on the possible resulting prism manifolds, and also prove that $4m$ and $4m+4$ are characterizing slopes for the torus knot $T(2m+1,2)$ for each $m\geq 1$.

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BibTeX RIS

Yi Ni, Xingru Zhang. 2016-07-19. Finite Dehn surgeries on knots in $S^3$. https://doi.org/10.2140/agt.2018.18.441

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