Renormalization matching algebraic and combinatorial web evaluations

Determine whether a renormalization of the algebraic evaluation of type \(\mathsf{D}\) webs exists that matches the combinatorially defined web evaluation, and clarify the role of pivotal structures in such a renormalization.

Background

The paper constructs a combinatorial state-sum evaluation for type D\mathsf{D} webs and relates the resulting framed-link invariant to the Reshetikhin–Turaev invariant associated with the vector representation of Uq(soN)U_q(\mathfrak{so}_N), after replacing qq by −q-q. Thus, the link invariants agree only after this parameter substitution.

The authors explicitly leave unresolved whether the algebraic evaluation of webs itself can be renormalized so that it agrees directly with the combinatorial evaluation. They suggest that examining pivotal structures may illuminate the existence and form of such a renormalization.

References

We did not prove that there is a re-normalization of the algebraic evaluation of webs which matches the combinatorial one. We believe that considering pivotal structures should shed some light on whether there is such a renormalization.

— MOY calculus in type D  (2501.04332 - Bodish et al., 8 Jan 2025) in Final Remark in Section 7, “Type \(\mathsf{D}\) intertwiners,” following Theorem \ref{thm:ev-vs-alg}