Connections between Floer-type invariants and Morse-type invariants of Legendrian knots.

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.

Name thesis:degree_name

Doctor of Philosophy (PhD)

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Year dc:date.available

2009

Author dc:creator

• Henry, Michael

Contributors dc:contributor

• Rachel Roberts

• Mathematics
• contact topology
• knot theory
• Legendrian knot theory
• Low-dimensional topology
• Topology

Language dc:language

English (en)

DOI

https://doi.org/10.7936/K71G0JBG

OAI identifier oai:identifier

oai:openscholarship.wustl.edu:etd-1146