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Transactions of the American Mathematical Society

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

 
 

 

Transverse Surgery on Knots in Contact 3-Manifolds


Author: James Conway
Journal: Trans. Amer. Math. Soc. 372 (2019), 1671-1707
MSC (2010): Primary 53D10, 57R17
DOI: https://doi.org/10.1090/tran/7611
Published electronically: May 9, 2019
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Abstract: We study the effect of surgery on transverse knots in contact 3-manifolds. In particular, we investigate the effect of such surgery on open books, the Heegaard Floer contact invariant, and tightness. One main aim of this paper is to show that in many contexts, transverse surgery is a more natural tool than surgery on Legendrian knots.

We reinterpret contact $ (\pm 1)$-surgery on Legendrian knots as transverse surgery on transverse push-offs, allowing us to give simpler proofs of known results. We give the first result on the tightness of inadmissible transverse surgery (cf. contact $ (+1)$-surgery) for contact manifolds with vanishing Heegaard Floer contact invariant. In particular, inadmissible transverse $ r$-surgery on the connected binding of a genus $ g$ open book that supports a tight contact structure preserves tightness if $ r > 2g-1$.


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Additional Information

James Conway
Affiliation: Department of Mathematics, University of California, Berkeley, Berkeley, California 94720
Email: conway@berkeley.edu

DOI: https://doi.org/10.1090/tran/7611
Received by editor(s): March 14, 2018
Received by editor(s) in revised form: May 3, 2018
Published electronically: May 9, 2019
Additional Notes: The author was partially supported by NSF Grant DMS-13909073.
Article copyright: © Copyright 2019 American Mathematical Society