The $Q$-polynomial is an invariant of the isotopy type of an unoriented link defined by Brandt, Lickorish, Millett, and Ho around 1985. It is shown that there exist infinitely many prime knots and links with trivial $Q$-polynomial, and so the $Q$-polynomial does not detect trivial links.
"Links with trivial $Q$-polynomial." J. Math. Soc. Japan 71 (1) 19 - 42, January, 2019. https://doi.org/10.2969/jmsj/77167716