Quantum one-cocycles for knots
arXiv:1304.0970
Abstract
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.
177 pages, 169 figures, an error corrected, references added
References in corpus (8)
- On the Hecke algebras and the colored HOMFLY polynomial
- Functional Tetrahedron Equation
- Integral Expressions for the Vassiliev Knot Invariants
- Topology of spaces of knots in dimension 3
- Quantum Knot Invariants
- Knot polynomials via one parameter knot theory
- Isotopy invariants for closed braids and almost closed braids via loops in stratified spaces
- Calculus of the first non-trivial 1-cocycle of the space of long knots