论文标题
打结的ADO和彩色琼斯多项式的统一
A unification of the ADO and colored Jones polynomials of a knot
论文作者
论文摘要
在本文中，我们证明了$ s^3 $的有色琼斯多项式的家族决定了这个结的ado多项式家族。更确切地说，我们构建了两个变量结，统一了ADO和彩色琼斯多项式。一方面，可以以$ 2R $ unity的$ 2R $ r \ in \ bbb n^*$进行$ 2R $ unity的根源评估，我们可以通过亚历山大多项式获得ADO多项式。另一方面，在$ a = q^n $中评估的第二个变量$ a $提供了彩色的琼斯多项式。由此，我们展示了一张地图发送，对于任何结，有色琼斯的家族多项式向Ado多项式家庭提供了多项式。作为这一事实的直接应用，我们将证明每个ADO多项式都是Q-Onolonomic的，并且与彩色Jones功能相同的多项式歼灭。统一不变的构建将使用戒指和代数的完成。我们还将展示如何从Habiro的Quantum $ \ Mathfrak {SL} _2 $完成ARXIV研究：Math/0605313中恢复我们的不变性。
In this paper we prove that the family of colored Jones polynomials of a knot in $S^3$ determines the family of ADO polynomials of this knot. More precisely, we construct a two variables knot invariant unifying both the ADO and the colored Jones polynomials. On one hand, the first variable $q$ can be evaluated at $2r$ roots of unity with $r \in \Bbb N^*$ and we obtain the ADO polynomial over the Alexander polynomial. On the other hand, the second variable $A$ evaluated at $A=q^n$ gives the colored Jones polynomials. From this, we exhibit a map sending, for any knot, the family of colored Jones polynomials to the family of ADO polynomials. As a direct application of this fact, we will prove that every ADO polynomial is q-holonomic and is annihilated by the same polynomials as of the colored Jones function. The construction of the unified invariant will use completions of rings and algebra. We will also show how to recover our invariant from Habiro's quantum $\mathfrak{sl}_2$ completion studied in arXiv:math/0605313.
