The (2,1)-cable Γ-polynomials of knots up to ten crossings

Abstract

For coprime integers [Formula: see text] and [Formula: see text], the [Formula: see text]-cable [Formula: see text]-polynomial of a knot is the [Formula: see text]-polynomial of the [Formula: see text]-cable knot of the knot, where the [Formula: see text]-polynomial is the common zeroth coefficient polynomial of the HOMFLYPT and Kauffman polynomials. Since it is known that the [Formula: see text]-polynomial is computable in polynomial time, the [Formula: see text]-cable [Formula: see text]-polynomial is also computable in polynomial time. In this paper, we show that the [Formula: see text]-cable [Formula: see text]-polynomial completely classifies the unoriented knots up to ten crossings including the chirality information.