# A diophantine equation for the E8 knot polynomial family

Let $x,y,z$ be dimensions that appear in the Clebsch-Gordan series $x*x=1+t+u+y+z$. (E8 family if $t=x$ (say), but there is at least another family. E.g. B4(R4) belongs to the latter.)

With the right pick of dimension ($t,u,y,z$ are not equivalent!) I got the following diophantine equation:

\eqalign{-27(-1+x)x^2(2+x)^2y+54x(2+x)^2y^2+9(-1+x)x^2(2+x)^2z\cr-24(-4+x)(-1+x)x(2+x)yz +16(-1+x)(2+x)(8+x)y^2z\cr-18x^2(2+x)z^2+48(-4+x)xyz^2-32(8+x)y^2z^2=0\cr}

(No rational solutions please, as these are dimensions. I don't exclude negative $x,y,z$ for now, though.)
Solving after $y$, $$x(36x+36x^2+9x^3-32z+16xz+16x^2z-32z^2)=a^2$$ ($a$ is still integer) and solving that after $z$, $$2x^2(2+x)^3(1+2x)=2a^2x+b^2$$ ($b$ is also integer.)

I'm stuck here. Were this a hunt for rational solutions, I'd set $a=x(x+2)c$ and $b=x(x+2)d$ and use the standard method for Pythagorean triangles afterwards. But I'm not sure $c$ and $d$ are integers (even constraining to the actual solutions like E8(R1),...), and the Pythagorean parametrization usually also is done with rationals.

Can you still give a parametrization $x=f(p,q)$, $y=g(p,q)$, $z=h(p,q)$ with integer $p,q$? (As I said, with rational $p,q$ this is trivial, in fact I started from that!) This should be elementary number theory...but it's already too high for me.

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I'm a little confused and think there might be a typo somewhere. How do you get to something like Pythagorean triangles when there's an extra x floating around on the a^2 term? –  Noah Snyder Feb 9 '12 at 16:19
Tex, please. This is unreadable. –  Felipe Voloch Feb 9 '12 at 17:07
Putting things in tex is easy here, you just put dollar signs around the expressions, or double dollar signs if you want display equations (which you should for some of your larger expressions). Fortunately tex is very easy to use, and all of your expressions are already valid tex expressions. –  Noah Snyder Feb 9 '12 at 19:48
Also, one has to be careful when using the symbol "*" as it is typically used to italicize a word or a sentence. –  André Henriques Feb 9 '12 at 20:00
I have aTeXed the question. If I damaged anything, feel free to edit. –  Gerry Myerson Feb 9 '12 at 22:34

MR1960703 (2004g:17003a) Westbury, Bruce W. $R$-matrices and the magic square. J. Phys. A 36 (2003), no. 7, 1947--1959.