Mathematics – Geometric Topology
Scientific paper
2009-11-09
Mathematics
Geometric Topology
54 pages, 32 figures; v3. Definition of main object generalized slightly, other minor revisions, to appear in Pacific J. Math
Scientific paper
We define an algebraic/combinatorial object on the front projection $\Sigma$ of a Legendrian knot called a Morse complex sequence, abbreviated MCS. This object is motivated by the theory of generating families and provides new connections between generating families, normal rulings, and augmentations of the Chekanov-Eliashberg DGA. In particular, we place an equivalence relation on the set of MCSs on $\Sigma$ and construct a surjective map from the equivalence classes to the set of chain homotopy classes of augmentations of $L_\Sigma$, where $L_\Sigma$ is the Ng resolution of $\Sigma$. In the case of Legendrian knot classes admitting representatives with two-bridge front projections, this map is bijective. We also exhibit two standard forms for MCSs and give explicit algorithms for finding these forms.
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