The general link wavefunction conjecture

Let LL be a link with a well-defined GM series

FL(X1,,XN;q)=m12ZNfLm(q)Xm,F_L(X_1,\ldots,X_N;q)=\sum_{\mathbf m\in\frac12\mathbb{Z}^N}f^{\mathbf m}_L(q)X^{\mathbf m},

and let QQ be its linking matrix with zero diagonal. General link wavefunction conjecture. The wavefunction in H(T2)N\mathcal{H}(T^2)^{\otimes N} exists and is

L,a=nZN+aq(Qn,n)fLm(q)i=1NYiniXimi+(Qn)i.\lvert L,\mathbf a\rangle=\sum_{\mathbf n\in\mathbb{Z}^N+\mathbf a}q^{-(Q\mathbf n,\mathbf n)}f^{\mathbf m}_L(q)\prod_{i=1}^NY_i^{n_i}X_i^{m_i+(Q\mathbf n)_i}.

The claim extends the explicitly derived plumbed-link formula to arbitrary links with well-defined GM series; convergence and the general existence statement are not established in the source.

Sources & referencesView supporting material

Primary source

Pedro Guicardi and Mrunmay Jagadale, “Z-TQFT, Surgery Formulas, and New Algebras”, arXiv:2509.14311 (2025).

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