Mathematics – Symplectic Geometry
Scientific paper
2010-04-28
J. Topology 4 (2011), no. 1, 73--104
Mathematics
Symplectic Geometry
Revised version, 38 pages, to appear in Journal of Topology
Scientific paper
10.1112/jtopol/jtq035
Given a front projection of a Legendrian knot $K$ in $\mathbb{R}^{3}$ which has been cut into several pieces along vertical lines, we assign a differential graded algebra to each piece and prove a van Kampen theorem describing the Chekanov-Eliashberg invariant of $K$ as a pushout of these algebras. We then use this theorem to construct maps between the invariants of Legendrian knots related by certain tangle replacements, and to describe the linearized contact homology of Legendrian Whitehead doubles. Other consequences include a Mayer-Vietoris sequence for linearized contact homology and a van Kampen theorem for the characteristic algebra of a Legendrian knot.
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