Satellite ruling polynomials, DGA representations, and the colored HOMFLY-PT polynomial
Caitlin Leverson
Georgia Institute of Technology, Atlanta, USADan Rutherford
Ball State University, Muncie, USA
Abstract
We establish relationships between two classes of invariants of Legendrian knots in : representation numbers of the Chekanov–Eliashberg DGA and satellite ruling polynomials. For positive permutation braids, , we give a precise formula in terms of representation numbers for the -graded ruling polynomial of the satellite of with specialized at with a prime power, and we use this formula to prove that arbitrary -graded satellite ruling polynomials, , are determined by the Chekanov–Eliashberg DGA of . Conversely, for , we introduce an -colored -graded ruling polynomial, , in strict analogy with the -colored HOMFLY-PT polynomial, and show that the total -dimensional -graded representation number of to , , is exactly equal to . In the case of \nbdash graded representations, we show that arises as a specialization of the -colored HOMFLY-PT polynomial.
Cite this article
Caitlin Leverson, Dan Rutherford, Satellite ruling polynomials, DGA representations, and the colored HOMFLY-PT polynomial. Quantum Topol. 11 (2020), no. 1, pp. 55–118
DOI 10.4171/QT/133