Formality Map
Universal Quantum Invariant
A universal quantum invariant, (a.k.a formality map, Vassiliev invariant) is a filtered homomorphism
such that . If such formality map exists, we call formal.
Theorem
Let
denote a filtered algebra, the associated graded algebra, and a candidate model for , with a surjective map . Assume that is a filtered homomorphism , such that . Then is an isomorphism, and is a universal quantum invariant.
Proof
Formality on Laurent Polynomials
Formality on Knots
Proposition
The algebra of knots,
is formal.
Proof Consider the the space of chord diagrams modulo the 1T and 4T relations,