Framed knot contact homology (Q2472322)

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 \textit{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.