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. In this paper, we show that there exist infinitely many knots with the trivial [Formula: see text]-cable [Formula: see text]-polynomial, that is, the [Formula: see text]-cable [Formula: see text]-polynomial of the trivial knot. Moreover, we see that the knots have the trivial [Formula: see text]-polynomial, the trivial first coefficient HOMFLYPT and Kauffman polynomials.
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