DRAFT
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Geometry and Topology Seminar

Friday, April 27, 2018
3:00pm to 5:00pm
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Building 15, Room 104
Legendrian satellite knots, DGA representations, and the colored HOMFLY-PT polynomial
Caitlin Leverson, School of Mathematics, Georgia Institute of Technology,
Legendrian knots are topological knots which satisfy extra geometric conditions. Two classes of invariants of Legendrian knots in $S^3$ are ruling polynomials and representations of the Chekanov-Eliashberg differential graded algebra (DGA). Given a knot $K$ and a positive permutation braid $\beta$, we give a precise formula relating a specialization of the ruling polynomial of the satellite $S(K,\beta)$ with certain counts of representations of the DGA of the original knot $K$. We also introduce an $n$-colored ruling polynomial, defined analogously to the $n$-colored HOMFLY-PT polynomial, and show that the 2-graded version of it arises as a specialization of the $n$-colored HOMFLY-PT polynomial. This is joint work with Dan Rutherford.
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