In this paper we show how to combinatorically compute the rotation class of a large family of embedded Legendrian tori in R 5 with the standard contact form. In particular, we give a formula to compute the Maslov index for any loop on the torus and compute the Maslov number of the Legendrian torus. These formulas are a necessary component in computing contact homology. Our methods use a new way to represent knotted Legendrian tori called Lagrangian hypercube diagrams.S. Baldridge was partially supported by NSF Grant DMS-0748636.
In this paper, we prove that the 2-factor polynomial, an invariant of a planar trivalent graph with a perfect matching, counts the number of 2factors that contain the the perfect matching as a subgraph. Consequently, we show that the polynomial detects even perfect matchings.
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. We will show that the cube number detects chirality in all cases computed thus far, and distinguishes certain legendrian knots.
In this paper, we prove that the $2$-factor polynomial, an invariant of a planar trivalent graph with a perfect matching, counts the number of $2$-factors that contain the perfect matching as a subgraph. Consequently, we show that the polynomial detects even perfect matchings.
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