Title page for ETD etd-05252012-130050

Type of Document Dissertation
Author McCarty, Ben
URN etd-05252012-130050
Title Hypercube diagrams for knots, links, and knotted tori
Degree Doctor of Philosophy (Ph.D.)
Department Mathematics
Advisory Committee
Advisor Name Title
Baldridge, Scott Committee Chair
Dasbach, Oliver Committee Member
Davidson, Mark Committee Member
Delzell, Charles Committee Member
Gilmer, Pat Committee Member
Van Scotter, James R Dean's Representative
  • knots
  • links
  • contact
  • Legendrian
Date of Defense 2012-05-01
Availability unrestricted
For a knot K the cube number is a knot invariant defined to be the smallest n for which there is a cube diagram of size n for K. Examples of knots for which the cube number detects chirality are presented. There is also a Legendrian version of this invariant called the Legendrian cube number. We will show that the Legendrian cube number distinguishes the Legendrian left hand torus knots with maximal Thurston-Bennequin number and maximal rotation number from the Legendrian left hand torus knots with maximal Thurston-Bennequin number and minimal rotation number.

Finally, there is a generalization of cube diagrams, called hypercube diagrams. We use such diagrams, which represent immersed Lagrangian tori in R^4 to study embedded Legendrian tori in the standard contact space. We then show how to

compute one of the classical invariants, the rotation class, and discuss applications to contact homology.

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