Mathematics

Prof. Louis H. Kauffman
University of Illinois at Chicago

Knots and links are topological types of embeddings of circles in three dimensional space.
This proposal is concerned with a very specific question: Does the Jones polynomial detect the unknot?
This question has been open since the Jones polynomial was discovered in 1983 and it received a particular focus in the form of the bracket state summation
model for the Jones polynomial.


We focus on the diagrammatic bracket state summation. The bracket state summation has been generalized
by Mikhail Khovanov to a homology theory for knots and links known as Khovanov homology. By arranging the bracket states of a knot or link diagram $K$ in the form of a category $Cat(K)$ where
the objects of the category are the bracket states and the generating morphisms are arrows between states that have one smoothing re-smoothed, Khovanov is able to make a homological measure of this category that is topologically invariant. In a nutshell, he constructs a functor $F$ from $Cat(K)$ to a Frobenius module category
and takes the cohomology of the category $Cat(K)$ with coefficients in the sheaf defined by the functor $F.$ Khovanov homology has been proven to detect the unknot by Kronheimer and
Mrowka. A graded Euler characteristic of the Khovanov homology reproduces the Jones polynomial.
There remains a huge gap between this stellar result of Kronheimer and Mrowka for Khovanov homology detecting the unknot, and the possiblity that the Jones polynomial itself detects the unknot.
Kronheimer and Mrowka succeed via gauge theory and a comparison of Khovanov homology with Instanton Floer Knot Homology.
It remains to be seen whether this comparison can be extended to the subtler comparison with the Jones polynomial itself.


We are investigating a generalization of the conjecture that the Jones polynomial detects the unknot.
A knotoid is an equivalence class of a planar knot diagram with two free ends. The ends are allowed to be in different regions of the diagram. Knotoid diagrams
are taken up to Reidemeister moves. The moves are not allowed to take an arc across either of the ends of the knotoid.
The bracket polynomial and hence the Jones polynomial can be immediately extended to knotoids. We conjecture that the Jones polynomial detects the unknotted knotoid.
This conjecture has the same level of plausibility as the original conjecture that the Jones polynomial detects the unknot. The knotoid conjecture generalizes the first conjecture because knotoids with their endpoints in the same region are equivalent to classical knots.