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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Null surgery on knots in L-spaces


Authors: Yi Ni and Faramarz Vafaee
Journal: Trans. Amer. Math. Soc. 372 (2019), 8279-8306
MSC (2010): Primary 57M27
DOI: https://doi.org/10.1090/tran/7510
Published electronically: September 23, 2019
MathSciNet review: 4029697
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Abstract: Let $ K$ be a knot in an L-space $ Y$ with a Dehn surgery to a surface bundle over $ S^1$. We prove that $ K$ is rationally fibered, that is, the knot complement admits a fibration over $ S^1$. As part of the proof, we show that if $ K\subset Y$ has a Dehn surgery to $ S^1 \times S^2$, then $ K$ is rationally fibered. In the case that $ K$ admits some $ S^1 \times S^2$ surgery, $ K$ is Floer simple, that is, the rank of $ \widehat {HFK}(Y,K)$ is equal to the order of $ H_1(Y)$. By combining the latter two facts, we deduce that the induced contact structure on the ambient manifold $ Y$ is tight.

In a different direction, we show that if $ K$ is a knot in an L-space $ Y$, then any Thurston norm minimizing rational Seifert surface for $ K$ extends to a Thurston norm minimizing surface in the manifold obtained by the null surgery on $ K$ (i.e., the unique surgery on $ K$ with $ b_1>0$).


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Additional Information

Yi Ni
Affiliation: Department of Mathematics, California Institute of Technology, Pasadena, California 91125
Email: yini@caltech.edu

Faramarz Vafaee
Affiliation: Department of Mathematics, California Institute of Technology, Pasadena, California 91125
Address at time of publication: Department of Mathematics, Duke University, Durham, North Carolina 27708
Email: vafaee@math.duke.edu

DOI: https://doi.org/10.1090/tran/7510
Received by editor(s): May 12, 2017
Received by editor(s) in revised form: January 9, 2018, and January 12, 2018
Published electronically: September 23, 2019
Additional Notes: The first author was partially supported by NSF grant numbers DMS-1103976, DMS-1252992, and an Alfred P. Sloan Research Fellowship.
The second author was partially supported by an NSF Simons travel grant.
Article copyright: © Copyright 2019 American Mathematical Society