DGA Representations, Ruling Polynomials, and the Colored HOMFLY-PT Polynomial

DGA Representations, Ruling Polynomials, and the Colored HOMFLY-PT Polynomial

Triangle Topology Seminar

Caitlin Leverson (Georgia Tech)

Monday, September 30, 2019 -
3:15pm to 4:15pm
Location: 
119 Physics

Given a pattern braid $\beta\in J^1(S^1)$, to any Legendrian knot $\Lambda$ in $\mathbb{R}^3$ with the standard contact structure, we can associate the Legendrian satellite knot $S(\Lambda,\beta)$. We will discuss the relationship between counts of augmentations of the Chekanov­Eliashberg differential graded algebra of $S(\Lambda,\beta)$ and counts of certain representations of the algebra of $\Lambda$. We will then define an $m$­graded $n$­colored ruling polynomial from the $m$­graded ruling polynomial, analogously to how the $n$­colored HOMFLY­PT polynomial is defined from the HOMFLY­PT polynomial, and extend results of the second author, to show that the $2$­graded $n$­colored ruling polynomial appears as a specialization of the $n$­colored HOMFLY­PT polynomial. This is joint work with Dan Rutherford.

Last updated: 2019/10/21 - 2:28pm