Abstract
We produce the first examples of closed, tight contact 3-manifolds which become overtwisted after performing admissible transverse surgeries. Along the way, we clarify the relationship between admissible transverse surgery and Legendrian surgery. We use this clarification to study a new invariant of transverse knots—namely, the range of slopes on which admissible transverse surgery preserves tightness—and to provide some new examples of knot types which are not uniformly thick. Our examples also illuminate several interesting new phenomena, including the existence of hyperbolic, universally tight contact 3-manifolds whose Heegaard Floer contact invariants vanish (and which are not weakly fillable); and the existence of open books with arbitrarily high fractional Dehn twist coefficients whose compatible contact structures are not deformations of co-orientable taut foliations.
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Acknowledgments
The authors thank Vincent Colin, Tobias Ekholm, Yasha Eliashberg, Ko Honda, Janko Latschev and Chris Wendl for helpful correspondence. JAB was partially supported by NSF Grant DMS-1104688 and JBE was partially supported by NSF Grant DMS-0804820 and thanks the University of Texas, Austin for its hospitality while working on parts of this paper.
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Baldwin, J.A., Etnyre, J.B. Admissible transverse surgery does not preserve tightness. Math. Ann. 357, 441–468 (2013). https://doi.org/10.1007/s00208-013-0911-8
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DOI: https://doi.org/10.1007/s00208-013-0911-8