A new symmetry of the colored Alexander polynomial

Abstract

We present a new conjectural symmetry of the colored Alexander polynomial, that is the specialization of the quantum ${\mathfrak{sl}}_N$ invariant widely known as the colored HOMFLY-PT polynomial. We provide arguments in support of the existence of the symmetry by studying the loop expansion and the character expansion of the colored HOMFLY-PT polynomial. We study the constraints this symmetry imposes on the group theoretic structure of the loop expansion and provide solutions to those constraints. The symmetry is a powerful tool for research on polynomial knot invariants and in the end we suggest several possible applications of the symmetry.

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Cited by in corpus (5)

• Chern-Simons perturbative series revisited
• Interplay between symmetries of quantum 6-j symbols and the eigenvalue hypothesis
• Evolution properties of the knot's defect
• A novel symmetry of colored HOMFLY polynomials coming from $\mathfrak{sl}(N|M)$ superalgebras
• Multistrand Eigenvalue conjecture and Racah symmetries