Perturbed Alexander Invariants via Quantum Cluster Algebras

Abstract

We derive a perturbative expansion of knot invariants using quantum cluster algebras. In a recent series of papers, it has been shown that the universal $R$-matrix of $U_q(\mathfrak{g})$, with $\mathfrak{g}$ a Lie algebra of type ADE, can be interpreted as a cluster transformation. For $\mathfrak{g} = \mathfrak{sl}_2$, we show that this interpretation yields a perturbed $R$-matrix. We achieve this by expressing certain representations of quantum cluster algebras in terms of generators of the Weyl-Heisenberg algebra. By inserting an auxiliary parameter $ε$, we derive a perturbed $R$-matrix. With it, we can compute a knot invariant whose zeroth-order term in $ε$ equals $ Δ_K (T)^{-1}$, the reciprocal of the Alexander polynomial, while higher-order terms in $ε$ give rise to so-called perturbed Alexander invariants. These invariants also appear in perturbative series known by several names in the literature: the Melvin-Morton-Rozansky (MMR) expansion, the large-color expansion, the rational expansion, or the loop expansion. We implement our approach formally in \textit{Mathematica} and compute explicit examples. Moreover, we argue that the methods developed in this paper should extend beyond the case $\mathfrak g=\mathfrak{sl}_2$.