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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
Supervisors
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
Knot Theory, and introduce several link invariants which can be roughly divided in these four categories:
combinatorial (arc and braid indexes), geometric (canonical and slice genus), polynomial (Jones,
HOMFLY-PT and Kauffman polynomials) and homological invariants (Khovanov Homology).
The second chapter is dedicated to the study of Contact Structures on 3-manifolds, Legendrian/transversal
isotopy classes of contact links and their invariants. In particular we define the classical invariants for
these links and show how to compute them from a front diagram.
The previously mentioned result of Bennequin provides an upper bound in terms of topological prop-
erties of the link for two of these invariants, namely the Thurston-Bennequin number and the Self Link-
ing number in the Legendrian and transverse case respectively. These two integers basically measure the
signed rotations of the contact planes around a link with respect to a fixed framing.
The final chapter combines the previous ones: we define the Maximal Thurston-Bennequin (respec-
tively Maximal Self Linking) number TB(L) of a topological link L as the maximum among all Thurston-
Bennequin numbers of Legendrian (respectively transversal) links belonging to the class of L. We show
that these new topological invariants can be bounded above by expressions that depend on some topolog-
ical property of the link (usually the minimum degree of a polynomial invariant). In particular we give
and use a bound recently proved by Lenhard Ng involving Khovanov Homology.
We conclude by examining the sharpness of these bounds for tb on certain families of knots, and by
generalizing a Theorem of Ng regarding the Thurston-Bennequin invariant of a doubled knot.
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