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Tesi etd-05282012-113646


Thesis type
Tesi di laurea magistrale
Author
CELORIA, DANIELE
URN
etd-05282012-113646
Title
Relations among Topological and Contact Knot Invariants
Struttura
SCIENZE MATEMATICHE, FISICHE E NATURALI
Corso di studi
MATEMATICA
Commissione
relatore Lisca, Paolo
controrelatore Martelli, Bruno
Parole chiave
  • geometria di contatto
  • bassa dimensione
  • omologia di khovanov
  • invariante di Thurston-Bennequin
  • teoria dei nodi
Data inizio appello
15/06/2012;
Consultabilità
completa
Riassunto analitico
The thesis is structured as follows. In the first chapter we are going to give the basic notions of<br>Knot Theory, and introduce several link invariants which can be roughly divided in these four categories:<br> combinatorial (arc and braid indexes), geometric (canonical and slice genus), polynomial (Jones,<br>HOMFLY-PT and Kauffman polynomials) and homological invariants (Khovanov Homology).<br>The second chapter is dedicated to the study of Contact Structures on 3-manifolds, Legendrian/transversal<br>isotopy classes of contact links and their invariants. In particular we define the classical invariants for<br>these links and show how to compute them from a front diagram.<br>The previously mentioned result of Bennequin provides an upper bound in terms of topological prop-<br>erties of the link for two of these invariants, namely the Thurston-Bennequin number and the Self Link-<br>ing number in the Legendrian and transverse case respectively. These two integers basically measure the<br>signed rotations of the contact planes around a link with respect to a fixed framing.<br>The final chapter combines the previous ones: we define the Maximal Thurston-Bennequin (respec-<br>tively Maximal Self Linking) number TB(L) of a topological link L as the maximum among all Thurston-<br>Bennequin numbers of Legendrian (respectively transversal) links belonging to the class of L. We show<br>that these new topological invariants can be bounded above by expressions that depend on some topolog-<br>ical property of the link (usually the minimum degree of a polynomial invariant). In particular we give<br>and use a bound recently proved by Lenhard Ng involving Khovanov Homology.<br>We conclude by examining the sharpness of these bounds for tb on certain families of knots, and by<br>generalizing a Theorem of Ng regarding the Thurston-Bennequin invariant of a doubled knot.
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