Abstract.We introduce an infinite collection of (Laurent) polynomials associated with a 2-bridge knot or link normal form K = (a, ß). Experimental evidence suggests that these "Heckoid polynomials" define the affine representation variety of certain groups, the Heckoid groups, for K . We discuss relations which hold in the image of the generic representation for each polynomial. We show that, with a certain change of variable, each Heckoid polynomial divides the nonabelian representation polynomial of L , where L belongs to an infinite collection of 2-bridge knots/links determined by K and the Heckoid polynomial. Finally, we introduce a "precusp polynomial" for each 2-bridge knot normal form, and show it is the product of two (possibly reducible) nonconstant polynomials. We are preparing a sequel on the Heckoid groups and the evidence for some of the geometrical assertions stated in the introduction.A Heckoid group Y c SL2(C) is a Kleinian group (of second kind, acting discontinuously on an open set of the extended complex plane) which contains elliptics and is generated by parabolics in a particular way: there is a parabolic P £ r and an elliptic S £ SL2(C) such that T = (S^PS'" : v £ Z). All known examples of such groups are geometrically finite, but we shall also require geometric finiteness just to be on the safe side. The classical Hecke groups are essentially the first examples of Heckoid groups, the caveat being that E. Hecke took (P, S) as his group, whereas for us S only sometimes belongs to its Heckoid group. The "modest example" of [6] is an even Heckoid group for the pretzel knot (3,3,3), and in [1] M. Grayson showed that it gives a hyperbolic orbifold structure of infinite volume to the knot complement S* -(3, 3, 3).This sort of thing is what the present paper is leading up to, but is not actually about.We shall mainly be studying (Laurent) polynomials of various sorts associated with nonabelian representations in SL2(C) of a 2-bridge knot or link group, nK. This group is a 2-generator one relator group, nK = \xi, x2 : reí |, and as preparation for representing it we substitute matrices C, D for xx, x2 in rel, where C, D £ SL2(.R) are in normal form over a suitable (Laurent) polynomial algebra R over Z. Then rel(C, D) = E is equivalent to four polynomial equations in R, but in our 2-bridge case only one is needed, say Ojf = 0. In [7] we studied the polynomial O* for a 2-bridge knot and determined many of its properties. In § 1 below we review this material and add one new property that we found too late for [7]. However the primary purpose of § 1 is to switch from the nonunimodular normal form C, D used in [7] to a unimodular normal
