Satellite ruling polynomials, DGA representations, and the colored HOMFLY-PT polynomial

  • Caitlin Leverson

    Georgia Institute of Technology, Atlanta, USA
  • Dan Rutherford

    Ball State University, Muncie, USA
Satellite ruling polynomials, DGA representations, and the colored HOMFLY-PT polynomial cover

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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