A complete knot invariant from contact homology

Tobias Ekholm Uppsala University Lenhard Ng Duke University Vivek Shende University of California

Symplectic Geometry mathscidoc:1811.43005

Distinguished Paper Award in 2018

Inventiones Mathematicae, 2018
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, holomorphiccurve 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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  title={A complete knot invariant from contact homology},
  author={Tobias Ekholm, Lenhard Ng, and Vivek Shende},
  booktitle={Inventiones Mathematicae},
Tobias Ekholm, Lenhard Ng, and Vivek Shende. A complete knot invariant from contact homology. 2018. In Inventiones Mathematicae. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20181126100007595076182.
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