论文标题
亚历山大多项式作为通用不变
The Alexander polynomial as a universal invariant
论文作者
论文摘要
令$ \ mathsf {b} _1 $为多项式环$ \ mathbb {c} [a^{\ pm1}，b] $，与复杂的hopf代数的结构是由其解释为定期功能的代数所引起的复杂型号的代数，其属性属性属于复杂的trianian $ trian-trian-trian $ trian $ trian $ triangull $ 2-均为$ 2- \ begin {smallmatrix} a＆b \\ 0＆1 \ end {smallmatrix} \ right）$。我们证明，与$ \ Mathsf {b} _1 $相关的长结$ k $的通用不变性是规范归一化的亚历山大多项式$Δ_K（a）$的倒数。鉴于$ \ mathsf {b} _1 $承认$ q $ -deformation $ \ mathsf {b} _q $是基于（彩色的）琼斯多项式的基础，我们的结果为Melvin--Morton--Morton-Morton--Morton--Morton--Morton-Morton-Morton--Morton-Morton--Morton-Morton-Morton-Morton-rozansky猜想提供了bar-Nathanan and GarulIdis and garoufffalis和garoufff，
Let $\mathsf{B}_1$ be the polynomial ring $\mathbb{C}[a^{\pm1},b]$ with the structure of a complex Hopf algebra induced from its interpretation as the algebra of regular functions on the affine linear algebraic group of complex invertible upper triangular 2-by-2 matrices of the form $\left( \begin{smallmatrix} a&b\\0&1 \end{smallmatrix}\right)$. We prove that the universal invariant of a long knot $K$ associated to $\mathsf{B}_1$ is the reciprocal of the canonically normalised Alexander polynomial $Δ_K(a)$. Given the fact that $\mathsf{B}_1$ admits a $q$-deformation $\mathsf{B}_q$ which underlies the (coloured) Jones polynomials, our result provides another conceptual interpretation for the Melvin--Morton--Rozansky conjecture proven by Bar-Nathan and Garoufalidis, and Garoufalidis and Lê.
