Null surgery on knots in L-spaces
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- by Yi Ni and Faramarz Vafaee PDF
- Trans. Amer. Math. Soc. 372 (2019), 8279-8306 Request permission
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
- MR Author ID: 737604
- 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
- MR Author ID: 1093993
- Email: vafaee@math.duke.edu
- 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. - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 8279-8306
- MSC (2010): Primary 57M27
- DOI: https://doi.org/10.1090/tran/7510
- MathSciNet review: 4029697