A perturbative expansion of knot invariants is derived using quantum cluster algebras. By interpreting the $R$-matrix of $U_q(\mathfrak{ sl}_2)$ as a cluster transformation and introducing an auxiliary parameter $\epsilon$, we derive a perturbed $R$-matrix expressed in terms of Heisenberg algebra generators arising from the representation theory of the quantum cluster algebra. The resulting knot invariant has a zeroth-order term equal to $\Delta_K(T)^{ -1}$, the reciprocal of the Alexander polynomial, while higher-order terms in $\epsilon$ produce perturbed Alexander polynomials. The construction combines the Schr\"odinger representation of the quantum torus algebra with cluster mutation combinatorics and is illustrated with a Mathematica implementation and explicit examples.
Categories: math.GT