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THE CONORMAL TORUS IS A COMPLETE KNOT INVARIANT

Part of: Homology and cohomology theories Low-dimensional topology

Published online by Cambridge University Press:  06 September 2019

VIVEK SHENDE Open the ORCID record for VIVEK SHENDE [Opens in a new window]
VIVEK SHENDE*
Affiliation:
Departments of Mathematics, U.C. Berkeley, Berkeley CA 94720, USA; [email protected]

Abstract

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We use microlocal sheaf theory to show that knots can only have Legendrian isotopic conormal tori if they themselves are isotopic or mirror images.

MSC classification

Primary: 57M25: Knots and links in $S^3$ 57M27: Invariants of knots and 3-manifolds 55N30: Sheaf cohomology
Type
Research Article
Information
Forum of Mathematics, Pi , Volume 7 , 2019 , e6
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike licence (http://creativecommons.org/licenses/by-nc-sa/4.0/), which permits noncommercial re-use, distribution, and reproduction in any medium, provided the same Creative Commons licence is included and the original work is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use.
Copyright
© The Author 2019

References

Abouzaid, M., ‘Nearby Lagrangians with vanishing Maslov class are homotopy equivalent’, Invent. Math. 189(2) (2012), 251–313.Google Scholar
Abouzaid, M. and Kragh, T., ‘Simple homotopy equivalence of nearby Lagrangians’, Acta Mathematica 220(2) (2018), 207–237.Google Scholar
Beilinson, A., Bernstein, J. and Deligne, P., ‘Faisceaux pervers’, Astérisque 100 (1982), 5–171.Google Scholar
Berest, Y., Eshmatov, A. and Yeung, W.-K., ‘Perverse sheaves and knot contact homology’, Comptes Rendus Mathematique 355(4) (2017), 378–399.Google Scholar
Bondal, A. and Orlov, D., ‘Reconstruction of a variety from the derived category and groups of autoequivalences’, Compositio Math. 125(3) (2001), 327–344.Google Scholar
Bott, R., ‘On manifolds all of whose geodesics are closed’, Ann. of Math. (2) 60 (1954), 375–382.Google Scholar
Chiu, S.-F., ‘Non-squeezing property of contact balls’, Duke Mathematical Journal 166(4) (2017), 605–655.Google Scholar
Cieliebak, K., Ekholm, T., Latschev, J. and Ng, L., ‘Knot contact homology, string topology, and the cord algebra’, Journal de l’École polytechnique-Mathématiques 4 (2017), 661–780.Google Scholar
Cornwell, C. R., ‘KCH representations, augmentations, and A-polynomials’, Journal of Symplectic Geometry 15(4) (2017), 983–1017.Google Scholar
Cornwell, C. R., ‘Knot contact homology and representations of knot groups’, J. Topol. 7(4) (2014), 1221–1242.Google Scholar
Ekholm, T., Ng, L. and Shende, V., ‘A complete knot invariant from contact homology’, Inventiones mathematicae 211(3) (2018), 1149–1200.Google Scholar
Fukaya, K., Seidel, P. and Smith, I., ‘Exact Lagrangian submanifolds in simply-connected cotangent bundles’, Invent. Math. 172(1) (2008), 1–27.Google Scholar
Gray, J., ‘Some global properties of Contact structures’, Ann. of Math. (2) 69(2) (1959), 421–450.Google Scholar
Gordon, C. and Luecke, J., ‘Knots are determined by their complements’, J. Amer. Math. Soc. 2(2) (1989), 371–415.Google Scholar
Gordon, C. and Lidman, T., ‘Knot contact homology detects cabled, composite, and torus knots’, Proceedings of the American Mathematical Society 145(12) (2017), 5405–5412.Google Scholar
Guillermou, S., ‘Quantization of conic Lagrangian submanifolds of cotangent bundles’. Preprint, 2012, arXiv:1212.5818.Google Scholar
Guillermou, S., ‘The Gromov–Eliashberg theorem by microlocal sheaf theory’. Preprint, 2013, arXiv:1311.0187.Google Scholar
Guillermou, S., ‘The three cusps conjecture’. Preprint, 2016, arXiv:1603.07876.Google Scholar
Guillermou, S., Kashiwara, M. and Schapira, P., ‘Sheaf quantization of Hamiltonian isotopies and applications to nondisplaceability problems’, Duke Math. J. 161(2) (2012), 201–245.Google Scholar
Hertwick, M., ‘A counterexample to the isomorphism problem for integral group rings’, Ann. of Math. (2) 154(1) (2001), 115–138.Google Scholar
Higman, G., ‘The units of group-rings’, Proc. Lond. Math. Soc. (2) 46 (1940), 231–248.Google Scholar
Howie, J. and Short, H., ‘The band-sum problem’, J. Lond. Math. Soc. (2) 31(3) (1985), 571–576.Google Scholar
Kashiwara, M. and Schapira, P., Sheaves on Manifolds, Grundlehren Math. Wiss., 292 (Springer, 1990).Google Scholar
Nadler, D., ‘Microlocal branes are constructible sheaves’, Selecta Math. (N.S.) 15(4) (2009), 563–619.Google Scholar
Ng, L., ‘Framed knot contact homology’, Duke Math. J. 141(2) (2008), 365–406.Google Scholar
Orlov, D., ‘Derived categories of coherent sheaves and equivalences between them’, Russian Math. Surveys 58(3) (2003), 511–591.Google Scholar
Shende, V., Treumann, D. and Zaslow, E., ‘Legendrian knots and constructible sheaves’, Inventiones mathematicae 207(3) (2017), 1031–1133.Google Scholar
Shende, V., Treumann, D., Williams, H. and Zaslow, E., ‘Cluster varieties from Legendrian knots’. Duke Mathematical Journal, Preprint, 2015, arXiv:1512.08942, to appear.Google Scholar
Shende, V., Treumann, D. and Williams, H., ‘On the combinatorics of exact Lagrangian surfaces’. Preprint, 2016, arXiv:1603.07449.Google Scholar
Tamarkin, D., ‘Microlocal condition for non-displaceablility’, inAlgebraic and Analytic Microlocal Analysis (Springer, 2013), 99–223.Google Scholar
Waldhausen, F., ‘On irreducible 3-manifolds which are sufficiently large’, Ann. of Math. (2) 87(1) (1968), 56–88.Google Scholar