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.

我们提出了有色亚历山大多项式的新的猜想对称性，即量子\（\ mathfrak {sl} _N \）不变式的特化，被广泛称为有色HOMFLY-PT多项式。通过研究彩色HOMFLY-PT多项式的循环展开和字符展开，我们提供了支持对称性存在的论据。我们研究了这种对称性对循环展开的群理论结构施加的约束，并为这些约束提供了解决方案。对称性是研究多项式结不变式的有力工具，最后我们提出了对称性的几种可能应用。