Thomas Fiedler, Université Paul Sabatier
We introduce a new tool for distinguishing knots, which is potentially more powerful than knot invariants. Given a couple of long knot diagrams, for any “correspondence” of their crossings and for any auxiliary knot K, we associate to the couple a finite system of linear equations (by using “naively quantized” combinatorial 1-cocycles). The variables of the system do not depend on the choice of the auxiliary knot K. By varying K we obtain systems of an arbitrary high number of equations. If the couple of diagrams represent the same knot, then this extremely overdetermined linear system has a solution for at least one of the “correspondences” of crossings. The inverse is of course a very intriguing open question. If the couple represents the same knot, then our methode gives at least information about the Reidemeister III moves in regular isotopies of the two diagrams.