All Theses and Dissertations (ETDs)

Author's School

Graduate School of Arts & Sciences

Mathematics

English (en)

January 2009

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Rachel Roberts

Abstract

We investigate existing Legendrian knot invariants and discover new connections between the theory of generating families, normal rulings and the Chekanov-Eliashberg differential graded algebra: CE-DGA). Given a Legendrian knot $\sK$ with generic front projection $\sfront$, we define a combinatorial/algebraic object on $\sfront$ called a \emph{Morse complex sequence}, abbreviated MCS. An MCS encodes a finite sequence of Morse homology complexes. Every suitably generic generating family for $\sfront$ admits an MCS and every MCS has a naturally associated graded normal ruling. In addition, every MCS has a naturally associated augmentation of the CE-DGA of the Ng resolution $\sNgres$ of the front $\sfront$. In this manner, an MCS connects generating families, normal rulings and augmentations. We place an equivalence relation on the set $\sDMCS$ of MCSs on $\sfront$ and prove that there exists a natural surjection from the equivalence classes of $\sDMCS$, denoted $\sDMCSeq$, to the set of chain homotopy classes of augmentations of $\sNgres$, denoted $\sAugNgresch$. In the case of Legendrian isotopy classes admitting representatives with two-bridge front projections, $\sDMCSeq$ and $\sAugNgresch$ are in bijection.

DOI

https://doi.org/10.7936/K71G0JBG