In my S matrix classification attempts I encounter a lot of Dubrovnik polynomials of the form D(z-1/z,z^n) and D(-z+1/z,z^n). [Second variable is for writhe, n is an integer; for the first I don't relate to the skein relation because having a 50:50 chance of sign, I will in 100% choose the false :-)]
Anyway, the one I seek has n=1 and is very easily to describe: All knots evaluate to 1. (Still, e.g. it can detect the mirrors of 6_3_3, the smallest nonalternating link.) Some more values: unlink 2, Hopf z^2+1/z^2, 421 z^4+1/z^4, Whitehead 2
I wouldn't wonder if this specialization of the Dubrovnik polynomial (it's the "other" solution for dimension 2 S matrices besides the Jones) already has a special name. Does it?