One-Cocycles and Knot Invariants is about classical knots, i.e.,
smooth oriented knots in 3-space. It introduces discrete
combinatorial analysis in knot theory in order to solve a global
tetrahedron equation. This new technique is then used to construct
combinatorial 1-cocycles in a certain moduli space of knot
diagrams. The construction of the moduli space makes use of the
meridian and the longitude of the knot. The combinatorial
1-cocycles are therefore 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.
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