Legendrian satellite knots, DGA representations, and the colored HOMFLY-PT polynomial

Caitlin Leverson (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.