gn.general topology - SU(2) representations of alternating knot groups

This looks like a good place to start from (if you haven't already read it)

MR2488756 (2009m:57024) Nagasato, Fumikazu . Finiteness of a section of the ${rm SL}(2,Bbb C)$-character variety
of the knot group.
Kobe J. Math. 24 (2007), no. 2, 125--136.

This paper shows that for any knot, there are only finitely many irreducible metabelian characters in the ${rm SL}(2,{bf C})$ character variety. It is also shown that the number of conjugacy classes is given by a simple formula involving the Alexander polynomial. In the context of two bridge knots, there are inequalities involving the $A$-polynomial of the knot. Results of this nature were previously obtained by X. S. Lin [Acta Math. Sin. (Engl. Ser.) 17 (2001), no. 3, 361--380; MR1852950 (2003f:57018)] for ${rm SU}(2)$ representations.

This paper can be found at www.math.titech.ac.jp/~fukky/metabelian.pdf