Re: Special Kauffman polynome values

In news:<[EMAIL PROTECTED]> schrieb Hauke Reddmann:
> I wonder whether there are "interesting" parameter values
> for the Kauffman polynome K(A) (with the writhe variable set
> to 1 as the most basic invariants should be achiral). 
> E.g. A=golden mean
> All knots/links have the value +-sqrt(5)^n
> A=-2
    ^ Maybe the sign is wrong?!
> All knots/links are integer squares
> 
> Surely this has been investigated?

I guess, your version of Kauffman polynomial K(A) 
is what other people call the Brandt-Lickorish-Millett-Ho polynomial 
(or sometimes the $Q$-polynomial) 
        Q_L(x) := F_L(1,x)

(see 
        http://mathworld.wolfram.com/BLMHoPolynomial.html
        http://mathworld.wolfram.com/KauffmanPolynomialF.html )
 
In this case you can find what you want in the original paper 

        Brandt, R. D.; Lickorish, W. B. R.; and Millett, K. C. 
        "A Polynomial Invariant for Unoriented Knots and Links." 
        Invent. Math. 84, 563-573, 1986

where amongst other properties you find that 
        Q(1)=1
        Q(-1)=(-3)^d where d is the dimension of the mod 3 homology 
                             of the 2-fold branched cover of L
        Q(2)=delta_L^2 where delta_L is the determinant of L 
and     Q(-2)=(-1)^{c-1} where c s the number of components of L 

The value at the golden mean follows of course from the more general, 
simple formula for the Kauffman polynomial of disjoint unions 

        F_{L1 cup L2} =[(a^{-1} + a)x^{-1} - 1] F_L1 F_L2 


The following two articles might also be of interest to you: 

        Jones, V. F. R. 
        On a certain value of the Kauffman polynomial.  
        Comm. Math. Phys.  125  (1989),  no. 3, 459--467.

        Stoimenow, A.
        Branched cover homology and $Q$ evaluations.
        Osaka J. Math. 39 (2002), no. 1, 13--21.
                                                                                
Regards, 
Thomas