A New Symmetry of the Colored Alexander Polynomial

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.