Closed Formulas for $η$-Corrections in the Once Punctured Torus
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866914560214040576 |
|---|---|
| author | Vargas, Nelson A. Colon |
| author_facet | Vargas, Nelson A. Colon |
| contents | We study $η$-correction terms in the Kauffman bracket skein algebra of the once-punctured torus $K_t(Σ_{1,1})$. 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 $(η)$. 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 $ε_n$ 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 $η$-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 $K_t(Σ_{1,1})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_18334 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Closed Formulas for $η$-Corrections in the Once Punctured Torus Vargas, Nelson A. Colon Geometric Topology Quantum Algebra We study $η$-correction terms in the Kauffman bracket skein algebra of the once-punctured torus $K_t(Σ_{1,1})$. 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 $(η)$. 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 $ε_n$ 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 $η$-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 $K_t(Σ_{1,1})$. |
| title | Closed Formulas for $η$-Corrections in the Once Punctured Torus |
| topic | Geometric Topology Quantum Algebra |
| url | https://arxiv.org/abs/2508.18334 |