Author's Department/Program
Mathematics
Language
English (en)
Date of Award
January 2009
Degree Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Chair and Committee
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.
Recommended Citation
Henry, Michael, "Connections between Floer-type invariants and Morse-type invariants of Legendrian knots." (2009). All Theses and Dissertations (ETDs). 147.
https://openscholarship.wustl.edu/etd/147
Comments
Permanent URL: http://dx.doi.org/10.7936/K71G0JBG