73纽结及其镜像的着色Jones多项式的尾部
The Tails of the Colored Jones Polynomials for 73 Knot and Its Mirror Image
DOI: 10.12677/aam.2026.157318, PDF,    科研立项经费支持
作者: 田金鹭, 李 佳, 冷旭东*：辽宁师范大学数学学院，辽宁 大连
关键词: 着色Jones多项式；尾部；q-级数；Temperley-Lieb代数；Colored Jones Polynomial； Tail； q-Series； Temperley-Lieb Algebra
摘要: 交错纽结着色Jones多项式具有系数渐进稳定性的特性，其零阶稳定性对应着色Jones多项式的尾部。现有研究表明，纽结着色Jones多项式的尾部能够自然对应Rogers-Ramanujan恒等式、伪theta函数等经典组合恒等式。本文以73纽结及其镜像为例，通过Temperley-Lieb代数中的气泡展开公式(bubble expansion formula)直接计算出着色Jones多项式的尾部，进而利用数论技巧证明该结果与Garoufalidis等人的猜想表达式一致。
Abstract: The colored Jones polynomial of an alternating knot exhibits the property of coefficient asymptotic stability, where zero-order stability corresponds to the tail of the colored Jones polynomial. Existing research indicates that the tail of the colored Jones polynomial for knots naturally corresponds to classical number theoretic identities such as the Rogers-Ramanujan identities and false theta functions. This paper takes 73 knots and it mirror image as examples, directly computes the tail of the colored Jones polynomial using the bubble expansion formula from the Temperley-Lieb algebra, and then employs number-theoretic techniques to prove that this result is consistent with the conjectured expression by Garoufalidis et al.
文章引用：田金鹭, 李佳, 冷旭东. 73纽结及其镜像的着色Jones多项式的尾部[J]. 应用数学进展, 2026, 15(7): 229-240. https://doi.org/10.12677/aam.2026.157318

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