One-Cocycles and Knot Invariants

One-Cocycles and Knot Invariants is about classical knots, i.e. smooth oriented knots in three-space. It introduces discrete combinatorial analysis in knot theory in order to solve a global tetrahedron equation. This new technique is then used in order to construct combinatorial one-cocycles in a certain moduli space of knot diagrams. The construction of the moduli space makes use of the meridian and of the longitude of the knot. The combinatorial 1-cocycles are then lifts of the well-known Conway polynomial of knots and they can be calculated in polynomial time. The 1-cocycles can distinguish loops consisting of knot diagrams in the moduli space up to homology. They give knot invariants when they are evaluated on canonical loops in the connected components of the moduli space. They are a first candidate for numerical knot invariants which can perhaps distinguish the orientation of knots.