Abstract
We construct an enhanced version of knot contact homology, and show that we can deduce from it the group ring of the knot group together with the peripheral subgroup. In particular, it completely determines a knot up to smooth isotopy. The enhancement consists of the (fully noncommutative) Legendrian contact homology associated to the union of the conormal torus of the knot and a disjoint cotangent fiber sphere, along with a product on a filtered part of this homology. As a corollary, we obtain a new, holomorphic-curve proof of a result of the third author that the Legendrian isotopy class of the conormal torus is a complete knot invariant.
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Notes
In fact, for the purposes of this paper and in particular the proof of Theorem 1.1, one could ignore orientations and work over \(\mathbb {Z}/2\) rather than \(\mathbb {Z}\). However, for the purposes of the general theory, we will work over \(\mathbb {Z}\) throughout.
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Acknowledgements
TE was supported by the Knut and Alice Wallenberg Foundation and the Swedish Research Council. LN was partially supported by NSF Grant DMS-1406371 and a Grant from the Simons Foundation (# 341289 to Lenhard Ng). VS was partially supported by NSF Grant DMS-1406871 and a Sloan Fellowship.
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Ekholm, T., Ng, L. & Shende, V. A complete knot invariant from contact homology. Invent. math. 211, 1149–1200 (2018). https://doi.org/10.1007/s00222-017-0761-1
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DOI: https://doi.org/10.1007/s00222-017-0761-1