What question did this study set out to answer?

The aim is to define and compute the Average Jones Polynomial (AJP) of oriented link shadows based on shadow graphs.

April 5, 2026Open Access

The Average Jones Polynomial: An Ensemble Approach to Knot Shadows

Key Points

The aim is to define and compute the Average Jones Polynomial (AJP) of oriented link shadows based on shadow graphs.
Defined the Average Jones Polynomial from over/under resolutions of link shadows.
Developed an algorithm to compute the AJP in O(2^n · poly(n)) time.
Analyzed the AJP's behavior under shadow Reidemeister moves.
The AJP is solely determined by the shadow graph, independent of knot diagrams.
The algorithm significantly improves performance compared to the brute-force method (from O(4^n) to O(2^n · poly(n))).
The AJP remains invariant under shadow Reidemeister 1 moves and transforms predictably under moves 2 and 3.

Abstract

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 (2ⁿ poly (n) ) time for a shadow with n crossings, which is a significant improvement compared to O (4ⁿ) 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.

The Average Jones Polynomial: An Ensemble Approach to Knot Shadows

Key Points

Abstract

Cite This Study