Abstract
A nontrivial slope r on a knot K in S3 is called a characterizing slope if whenever the result of r-surgery on a knot K’ is orientation-preservingly homeomorphic to the result of r-surgery on K, then K’ is isotopic to K. Ni and Zhang ask: for any hyperbolic knot K, is a slope r=p/q with |p|+|q| sufficiently large a characterizing slope? In this article, we prove that if we can take an unknot c so that (0,0)-surgery on K∪c results in S3 and c is not a meridian of K, then K has infinitely many noncharacterizing slopes. As the simplest known example, the hyperbolic, two-bridge knot 86 has no integral characterizing slopes. This answers the above question in the negative. We also prove that any L-space knot never admits such an unknot c.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 1461-1480 |
| Number of pages | 20 |
| Journal | Algebraic and Geometric Topology |
| Volume | 18 |
| Issue number | 3 |
| DOIs | |
| State | Published - Apr 9 2018 |
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Keywords
- Characterizing slope
- Dehn surgery
ASJC Scopus subject areas
- Geometry and Topology
Cite this
Noncharacterizing slopes for hyperbolic knots. / Baker, Kenneth; Motegi, Kimihiko.
In: Algebraic and Geometric Topology, Vol. 18, No. 3, 09.04.2018, p. 1461-1480.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Noncharacterizing slopes for hyperbolic knots
AU - Baker, Kenneth
AU - Motegi, Kimihiko
PY - 2018/4/9
Y1 - 2018/4/9
N2 - A nontrivial slope r on a knot K in S3 is called a characterizing slope if whenever the result of r-surgery on a knot K’ is orientation-preservingly homeomorphic to the result of r-surgery on K, then K’ is isotopic to K. Ni and Zhang ask: for any hyperbolic knot K, is a slope r=p/q with |p|+|q| sufficiently large a characterizing slope? In this article, we prove that if we can take an unknot c so that (0,0)-surgery on K∪c results in S3 and c is not a meridian of K, then K has infinitely many noncharacterizing slopes. As the simplest known example, the hyperbolic, two-bridge knot 86 has no integral characterizing slopes. This answers the above question in the negative. We also prove that any L-space knot never admits such an unknot c.
AB - A nontrivial slope r on a knot K in S3 is called a characterizing slope if whenever the result of r-surgery on a knot K’ is orientation-preservingly homeomorphic to the result of r-surgery on K, then K’ is isotopic to K. Ni and Zhang ask: for any hyperbolic knot K, is a slope r=p/q with |p|+|q| sufficiently large a characterizing slope? In this article, we prove that if we can take an unknot c so that (0,0)-surgery on K∪c results in S3 and c is not a meridian of K, then K has infinitely many noncharacterizing slopes. As the simplest known example, the hyperbolic, two-bridge knot 86 has no integral characterizing slopes. This answers the above question in the negative. We also prove that any L-space knot never admits such an unknot c.
KW - Characterizing slope
KW - Dehn surgery
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UR - http://www.scopus.com/inward/citedby.url?scp=85045224321&partnerID=8YFLogxK
U2 - 10.2140/agt.2018.18.1461
DO - 10.2140/agt.2018.18.1461
M3 - Article
AN - SCOPUS:85045224321
VL - 18
SP - 1461
EP - 1480
JO - Algebraic and Geometric Topology
JF - Algebraic and Geometric Topology
SN - 1472-2747
IS - 3
ER -