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 Keywords
Date of Defense 2012-05-01 Availability unrestricted AbstractFor 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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