Classical invariants of Legendrian knots in the 3-dimensional torus

All knots in R3 possess Seifert surfaces, and so the classical Thurston–Bennequin and rotation (or Maslov) invariants for Legendrian knots in a contact structure on R3 can be defined. The definitions extend easily to null-homologous knots in any 3-manifold M endowed with a contact structure ξ. We ge...

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Bibliographic Details
Published inTopology and its applications Vol. 185-186; pp. 65 - 79
Main Authors Schweitzer, Paul A., Souza, Fábio S.
Format Journal Article
LanguageEnglish
Published Elsevier B.V 01.05.2015
Subjects
3-Torus [formula omitted]
Contact structures
Legendrian knots
Maslov invariant
Seifert surfaces
Thurston–Bennequin invariant
Seifert surfaces
Contact structures
57R17
Legendrian knots
Thurston–Bennequin invariant
57M27
3-Torus T3
Maslov invariant
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Summary:All knots in R3 possess Seifert surfaces, and so the classical Thurston–Bennequin and rotation (or Maslov) invariants for Legendrian knots in a contact structure on R3 can be defined. The definitions extend easily to null-homologous knots in any 3-manifold M endowed with a contact structure ξ. We generalize the definition of Seifert surfaces and use them to define these invariants for all Legendrian knots, including those that are not null-homologous, in a contact structure on the 3-torus T3. We show how to compute the Thurston–Bennequin and rotation invariants in a tight oriented contact structure on T3 using projections.
ISSN:0166-8641
1879-3207
DOI:10.1016/j.topol.2015.02.006