Framed knot contact homology. (English) Zbl 1145.57010
The author defines new knot invariants over the ring \(\mathbb Z[\lambda^{\pm1},\mu^{\pm1}]\). The strongest invariant, and the one from which the others derive, is a differential graded algebra over \(\mathbb Z[\lambda^{\pm1},\mu^{\pm1}]\) \(l\), where \(l\) is a certain (geometric) equivalence relation. One of the several methods given for constructing the algebra is in terms of a closed braid representation of the knot. The connection with contact geometry is seen through this method. Three other methods are given, two of which use the “cords” of the author’s paper “Knot and braid invariants from contact homology II”, (with an appendix by the author and S. Gadgil) in [Geom. Topol. 9, 1603–1637 (2005; Zbl 1112.57001)]. Numerous properties and strengths of the invariants are discussed. For example the author proves that the Kinoshita-Terasaka knot and its Conway mutant are distinguished by their differential graded algebras.
Reviewer: Lee P. Neuwirth (Princeton)
MSC:
| 57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |
| 53D12 | Lagrangian submanifolds; Maslov index |
| 53D40 | Symplectic aspects of Floer homology and cohomology |
Citations:
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