The Average Jones Polynomial: An Ensemble Approach to Knot Shadows

This paper defines the Average Jones Polynomial (AJP) of an oriented link shadow as the uniform average of the Jones polynomials over all over/under resolutions of the shadow. This quantity is solely determined by the underlying shadow graph, independent of any particular knot diagram. It presents an algorithm that computes the AJP in $O(2^n \cdot \text{poly}(n))$ time for a shadow with $n$ crossings, which is a significant improvement compared to $O(4^n)$ complexity of the brute-force method. It then analyzes the behavior of the AJP under shadow Reidemeister moves, showing that it is invariant under the shadow Reidemeister 1 move and transforms in a predictable manner under the shadow Reidemeister 2 and 3 moves, indicating that the AJP captures structural information of the shadow.