Closed Formulas for $η$-Corrections in the Once Punctured Torus

Abstract: We study $\eta$-correction terms in the Kauffman bracket skein algebra of the once-punctured torus $K_t(\Sigma_{1,1})$. While the Frohman-Gelca product-to-sum rule is explicit on the closed torus, punctures introduce correction terms in the ideal $(\eta)$ whose structure has resisted systematic description beyond low-determinant cases. We give a closed form for the family $P_n=(1,0)\cdot T_n((1,2))$ (determinant $2$): the correction $\epsilon_n$ has an explicit Chebyshev expansion with coefficients that factor as geometric sums in $t^{\pm4}$ governed by parity. Tracking creation and cancellation of $\eta$ through the Chebyshev recurrence, and using diffeomorphism invariance, the formula transports to all products $C_1\cdot T_k(C_2)$ with $|\det(C_1,C_2)|=2$. We further treat the \emph{primitive maximal-thread} regime, where one Frohman-Gelca summand is fully threaded and the other is simple or doubly covered. In this case we obtain a closed form for the discrepancy: an $\eta$-linear cascade with Chebyshev $S$-coefficients that lowers the thread degree by two at each step. Equivalently, in this maximal-thread regime we solve the specialized Wang-Wong recursion in closed form; the coefficients match their fast algorithm term-by-term (up to the $T_0$ normalization) and subsume Cho's $|\det|=2$ case. The resulting rules give compact, scalable formulas for symbolic multiplication in $K_t(\Sigma_{1,1})$