  We give a method to construct non symmetric solutions of a global tetrahedron
equation from solutions of the Yang-Baxter equation. The solution in the
HOMFLYPT case gives rise to the first combinatorial quantum 1-cocycle which
represents a non trivial cohomology class in the topological moduli space of
long knots. We conjecture that the quotient of its values on Hatchers loop and
on the rotation around the long axis is related to the simplicial volume of the
knot complement in the 3-sphere and we prove this for the figure eight knot.
  Surprisingly, the formula for the solution in the HOMFLYPT case of the
positive global tetrahedron equation gives also a solution in the case of the
2-variable Kauffman invariant. But there is also a second solution giving rise
to yet another non trivial quantum 1-cocycle.
