2026-09-21T09:01:51Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/678142022-01-13T07:54:35Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Tomasz Mrowka.
Sivek, Steven (Steven W.)
Massachusetts Institute of Technology. Dept. of Mathematics.
Massachusetts Institute of Technology. Department of Mathematics
2011-12-19T19:00:59Z
2011-12-19T19:00:59Z
2011
2011
http://hdl.handle.net/1721.1/67814
767996443
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011.
Cataloged from PDF version of thesis.
Includes bibliographical references (p. 115-120).
In this thesis we apply techniques from the bordered and sutured variants of Floer homology to study Legendrian knots. First, given a front diagram for a Legendrian knot K in S₃ which has been split into several pieces, we associate a differential graded algebra to each "bordered" piece and prove a van Kampen theorem which recovers the Chekanov-Eliashberg invariant Ch(K) of the knot from the bordered DGAs. This leads to the construction of morphisms Ch(K) --> Ch(K') corresponding to certain Legendrian tangle replacements and many related applications. We also examine several examples in detail, including Legendrian Whitehead doubles and the first known knot with maximal Thurston-Bennequin invariant for which Ch(K) vanishes. Second, we use monopole Floer homology for sutured manifolds to construct new invariants of Legendrian knots. These invariants reside in monopole knot homology and closely resemble Heegaard Floer invariants due to Lisca-Ozsváth-Stipsicz-Szabó, but their construction directly involves the contact topology of the knot complement and so many of their properties are easier to prove in this context. In particular, we show that these new invariants are functorial under Lagrangian concordance.
by Steven Sivek.
Ph.D.
120 p.
eng
Massachusetts Institute of Technology
M.I.T. theses are protected by
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http://dspace.mit.edu/handle/1721.1/7582
Mathematics.
Bordered Legendrian knots and sutured Legendrian invariants
Thesis
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Publication
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<Abstract>In this thesis we apply techniques from the bordered and sutured variants of Floer homology to study Legendrian knots. First, given a front diagram for a Legendrian knot K in S₃ which has been split into several pieces, we associate a differential graded algebra to each "bordered" piece and prove a van Kampen theorem which recovers the Chekanov-Eliashberg invariant Ch(K) of the knot from the bordered DGAs. This leads to the construction of morphisms Ch(K) --> Ch(K') corresponding to certain Legendrian tangle replacements and many related applications. We also examine several examples in detail, including Legendrian Whitehead doubles and the first known knot with maximal Thurston-Bennequin invariant for which Ch(K) vanishes. Second, we use monopole Floer homology for sutured manifolds to construct new invariants of Legendrian knots. These invariants reside in monopole knot homology and closely resemble Heegaard Floer invariants due to Lisca-Ozsváth-Stipsicz-Szabó, but their construction directly involves the contact topology of the knot complement and so many of their properties are easier to prove in this context. In particular, we show that these new invariants are functorial under Lagrangian concordance.</Abstract>
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