Knot polynomials from 1-cocycles

Let M n be the topological moduli space of all parallel n-cables of long framed oriented knots in 3-space. We construct in a combinatorial way for each natural number n > 1 a 1-cocycle R n which represents a non trivial class in H 1 (M n ; Z[x 1 , x 2 , ..., x −1 1 , x −1 2 , ...]), where the number of variables x i depends on n. To each generic point in M n we associate in a canonical way an arc scan in M n , such that R n (scan) is already a polynomial knot invariant. We show that R 3 (scan) detects the non-invertibility of the knot 8 17 in a very simple way and without using the knot group. There are two well-known canonical loops in M n for each parallel n-cable of a long framed knot K: Gramain's loop rot and the Fox-Hatcher loop fh. The calculation of R n is of at most quartic complexity for these loops with respect to the number of crossings of K for each fixed n. It follows from results of Hatcher that K is not a torus knot if the rational function R n (f h(K))/R n (rot(K)) is not constant for each n > 1. ⊕ n R n is a natural candidate in order to separate all classes in H 1 (M 1 ; Q) ∼ = H 1 (M n ; Q), and in particular to distinguish all knot types π 0 (M 1)