| Résume | Skein theory gives both a method of producing knot invariants and a way to quantize character varieties of surfaces and the three manifolds they bound. There is a well supported but conjectural relationship between these two aspects of skeins -- that the character variety of a knot complement is determined in a specific way by the invariant of the corresponding knot. This is known as the AJ conjecture, because it was originally formulated as a relationship between two knot invariants known as the A-polynomial and the colored Jones polynomial. A missing ingredient in this conjecture is a concrete and general procedure for computing the quantized character variety of a knot complement, i.e. quantizing the A-polynomial.
The first half of this walk will be a friendly introduction to skein categories, algebras, and modules. The second half will introduce character varieties, explain their quantization by skein algebras, and discuss how this relates to the AJ conjecture. |
