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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?

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    $\begingroup$ This is probably one of the families SO(2n+1), SO(2n), SP(2n), or OSP(1|2n). Figuring out which requires chasing through a bunch of conventions, but if you look at the background to our paper (with Scott and Emily) arxiv.org/abs/1003.0022 you should be able to sort it out. In particular look at page 7 for our conventions for Dubrovnik and the bottom of page 9 for the specializations giving those families. $\endgroup$
    – Noah Snyder
    Commented Apr 20, 2011 at 16:37
  • $\begingroup$ Close enough. (Identified, but not "specially named".) The paper even answers some questions I didn't ask yet :-) $\endgroup$
    – Hauke Reddmann
    Commented Apr 21, 2011 at 13:50
  • $\begingroup$ Right, basically they only get special names if they were discovered and named prior to Reshetikhin-Turaev's work. Anyway which of those families did it turn out to be? I commented instead of answering just because I didn't know which one was the right answer, but it'd be good to actually have an answer posted. $\endgroup$
    – Noah Snyder
    Commented May 5, 2011 at 18:15

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