This talk introduces the Average Jones Polynomial (AJP), defined as the uniform expectation $V_{avg}(S, A) = 2^{-n} \sum_{D \in \mathcal{R}(S)} V_D(A)$, aimed to isolate the structural properties of the underlying 4-valent planar graph.

We model the shadow as an uncontracted Temperley-Lieb tensor network, $\mathcal{T}(S) = \prod_{i} (a\mathbf{1} + b e_i)$. This formulation reduces the computational complexity of AJP calculations to $O(n\alpha(n)2^n)$ and maps the AJP to a finite loop-model partition function $Z_S(\delta, a, b)$. This can be utilized to evaluate some macroscopic observables (e.g., the expected number of loops).

We analyze the behavior of AJP under shadow Reidemeister (SR) moves. The AJP is invariant under SR1 move, and transforms predictably under SR2 & SR3 moves. When evaluating $\Delta \mathcal{T} = \mathcal{T}(S’) - \mathcal{T}(S)$, SR2 and SR3 moves appear as $e_i$ and $(e_i - e_{i+1})$ defect terms, respectively. This suggests a lower bound of required SR moves to transform one shadow into another.