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Filtrations on the knot contact homology of transverse knots

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Abstract

We construct a new invariant of transverse links in the standard contact structure on \({\mathbb R }^3.\) This invariant is a doubly filtered version of the knot contact homology differential graded algebra (DGA) of the link, see (Ekholm et al., Knot contact homology, Arxiv:1109.1542, 2011; Ng, Duke Math J 141(2):365–406, 2008). Here the knot contact homology of a link in \({\mathbb R }^3\) is the Legendrian contact homology DGA of its conormal lift into the unit cotangent bundle \(S^*{\mathbb R }^3\) of \({\mathbb R }^3\), and the filtrations are constructed by counting intersections of the holomorphic disks of the DGA differential with two conormal lifts of the contact structure. We also present a combinatorial formula for the filtered DGA in terms of braid representatives of transverse links and apply it to show that the new invariant is independent of previously known invariants of transverse links.

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Notes

  1. Using the orientation on \(\partial D\) induced by the complex structure a puncture is positive if it maps the segment of the boundary just before the puncture to the lower sheet at the double point and the segment just after the puncture to the upper sheet. The puncture is negative if the roles of the upper and lower sheets are reversed.

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Authors and Affiliations

  1. Department of Mathematics, Uppsala University, Box 480, 751 06 , Uppsala, Sweden

    Tobias Ekholm

  2. Institut Mittag-Leffler, Aurav. 17, 182 60 , Djursholm, Sweden

    Tobias Ekholm

  3. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332-0160, USA

    John Etnyre

  4. Mathematics Department, Duke University, Durham, NC, 22708-0320, USA

    Lenhard Ng

  5. Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA, 01003-9305, USA

    Michael Sullivan

Authors
  1. Tobias Ekholm
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  2. John Etnyre
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  3. Lenhard Ng
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  4. Michael Sullivan
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Corresponding author

Correspondence to Michael Sullivan.

Additional information

The authors wish to thank MSRI for hosting them during this collaboration. TE was partially supported by the Göran Gustafsson Foundation for Research in Natural Sciences and Medicine. JBE was partially supported by NSF Grant DMS-0804820. LLN was partially supported by NSF Grant DMS-0706777 and NSF CAREER Grant DMS-0846346. MGS was partially supported by NSF Grants DMS-0707091 and DMS-1007260.

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Ekholm, T., Etnyre, J., Ng, L. et al. Filtrations on the knot contact homology of transverse knots. Math. Ann. 355, 1561–1591 (2013). https://doi.org/10.1007/s00208-012-0832-y

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  • DOI: https://doi.org/10.1007/s00208-012-0832-y

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