论文标题
在结，补充和6J符号上
On knots, complements, and 6j-symbols
论文作者
论文摘要
本文调查了彩色homfly-pt和kauffman同源性之间的关系，$ \ text {so}（n）$ Quantum $ 6J $ -Symbols和$（a，t）$ - 变形$ f_k $。首先，我们提出了一个简单的分级更改规则，它使我们能够从$ [r^2] $ - 彩色的四型四型homfly-ppt ppt the the Nots中获得$ [r] $ colored四级高级的曲棍球同源性。该规则源于表示表示的同构$（\ mathfrak {so} _6，[r]）\ cong（\ mathfrak {sl} _4，[r^2]）$。另外，我们发现$ $ - 多项式的SO和Su-Type的关系来自Kauffman同源性的差异。其次，我们提出了$ \ text {so}（n）（n \ geq 4）$ Quantum $ 6J $ -Symbols用于对称表示的封闭形式的表达，并计算当表示$ r = [1]，[2] $的情况时，计算情况的相应$ \ text {so}（so} so}（so} n）$ fusion矩阵。第三，我们猜测$（a，t）$ - 变形$ f_k $的封闭式表达式，用于带正辫子的双重扭结的补充。使用猜想的表达式，我们得出了$ t $变形的ADO多项式。
This paper investigates the relation between colored HOMFLY-PT and Kauffman homology, $\text{SO}(N)$ quantum $6j$-symbols and $(a,t)$-deformed $F_K$. First, we present a simple rule of grading change which allows us to obtain the $[r]$-colored quadruply-graded Kauffman homology from the $[r^2]$-colored quadruply-graded HOMFLY-PT homology for thin knots. This rule stems from the isomorphism of the representations $(\mathfrak{so}_6,[r]) \cong (\mathfrak{sl}_4,[r^2])$. Also, we find the relationship among $A$-polynomials of SO and SU-type coming from a differential on Kauffman homology. Second, we put forward a closed-form expression of $\text{SO}(N)(N\geq 4)$ quantum $6j$-symbols for symmetric representations, and calculate the corresponding $\text{SO}(N)$ fusion matrices for the cases when representations $R = [1],[2]$. Third, we conjecture closed-form expressions of $(a,t)$-deformed $F_K$ for the complements of double twist knots with positive braids. Using the conjectural expressions, we derive $t$-deformed ADO polynomials.
