Closed Formulas for $\eta$-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 gives an explicit multiplication formula on the closed torus, the once-punctured torus introduces correction terms in the ideal $(\eta)$. We give a closed formula for the Chebyshev-threaded family generated by the primitive determinant-two pair \[ P_n=T_n((1,2))\cdot(1,0). \] The correction $\epsilon_n$ has an explicit Chebyshev expansion whose coefficients factor as geometric sums in $t^{\pm4}$ and whose terms are governed by a parity pattern arising from the Chebyshev recurrence. We also treat a primitive maximal-thread regime, in which one Frohman--Gelca summand is fully threaded and the other is simple or doubly covered. In this case the discrepancy is an explicit $\eta$-linear cascade with Chebyshev $S$-coefficients, lowering the thread degree by two at each step. These formulas recover the relevant low-determinant behavior and give compact closed multiplication rules for structured threaded families in $K_t(\Sigma_{1,1})$.