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Knot contact homology detects cabled, composite, and torus knots


Authors: Cameron Gordon and Tye Lidman
Journal: Proc. Amer. Math. Soc. 145 (2017), 5405-5412
MSC (2010): Primary 57M25, 57M27, 57R17
DOI: https://doi.org/10.1090/proc/13643
Published electronically: June 16, 2017
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Abstract | References | Similar Articles | Additional Information

Abstract: Knot contact homology is an invariant of knots derived from Legendrian contact homology which has numerous connections to the knot group. We use basic properties of knot groups to prove that knot contact homology detects every torus knot. Further, if the knot contact homology of a knot is isomorphic to that of a cable (respectively composite) knot, then the knot is a cable (respectively composite).


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  • [BZ66] Gerhard Burde and Heiner Zieschang, Eine Kennzeichnung der Torusknoten, Math. Ann. 167 (1966), 169-176 (German). MR 0210113, https://doi.org/10.1007/BF01362170
  • [CELN16] K. Cieliebak, T. Ekholm, J. Latschev, and L. Ng, Relative contact homology, string topology, and the cord algebra, 2016, arXiv:1601.02167.
  • [Cor13] Christopher R. Cornwell, KCH representations, augmentations, and $ A$-polynomials, 2013, arXiv:1310.7526.
  • [EENS13] Tobias Ekholm, John B. Etnyre, Lenhard Ng, and Michael G. Sullivan, Knot contact homology, Geom. Topol. 17 (2013), no. 2, 975-1112. MR 3070519, https://doi.org/10.2140/gt.2013.17.975
  • [ENS16] Tobias Ekholm, Lenhard Ng, and Vivek Shende, A complete knot invariant from contact homology, 2016, arXiv:1606.07050.
  • [Hig40] Graham. Higman, The units of group-rings, Proc. London Math. Soc. (2) 46 (1940), 231-248. MR 0002137, https://doi.org/10.1112/plms/s2-46.1.231
  • [HS85] James Howie and Hamish Short, The band-sum problem, J. London Math. Soc. (2) 31 (1985), no. 3, 571-576. MR 812788, https://doi.org/10.1112/jlms/s2-31.3.571
  • [Ng05a] Lenhard Ng, Knot and braid invariants from contact homology. I, Geom. Topol. 9 (2005), 247-297. MR 2116316, https://doi.org/10.2140/gt.2005.9.247
  • [Ng05b] Lenhard Ng, Knot and braid invariants from contact homology. II, Geom. Topol. 9 (2005), 1603-1637. With an appendix by the author and Siddhartha Gadgil. MR 2175153, https://doi.org/10.2140/gt.2005.9.1603
  • [Ng08] Lenhard Ng, Framed knot contact homology, Duke Math. J. 141 (2008), no. 2, 365-406. MR 2376818, https://doi.org/10.1215/S0012-7094-08-14125-0
  • [Ng14] Lenhard Ng, A topological introduction to knot contact homology, Contact and symplectic topology, Bolyai Soc. Math. Stud., vol. 26, János Bolyai Math. Soc., Budapest, 2014, pp. 485-530. MR 3220948, https://doi.org/10.1007/978-3-319-02036-5_10
  • [She16] Vivek Shende, The conormal torus is a complete knot invariant, 2016, arXiv:1604.03520.
  • [Sim76] Jonathan Simon, Roots and centralizers of peripheral elements in knot groups, Math. Ann. 222 (1976), no. 3, 205-209. MR 0418079, https://doi.org/10.1007/BF01362577

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Additional Information

Cameron Gordon
Affiliation: Department of Mathematics, The University of Texas at Austin, Austin, TX, 78701, USA
Email: gordon@math.utexas.edu

Tye Lidman
Affiliation: Department of Mathematics, North Caroline State University, Raleigh, NC, 27603, USA
Email: tlild@math.ncsu.edu

DOI: https://doi.org/10.1090/proc/13643
Received by editor(s): November 3, 2015
Received by editor(s) in revised form: December 28, 2016
Published electronically: June 16, 2017
Additional Notes: The first author was partially supported by NSF Grant DMS-1309021. The second author was partially supported by NSF Grant DMS-1148490.
Communicated by: Kevin Whyte
Article copyright: © Copyright 2017 American Mathematical Society

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