论文标题
来自量子$ \ mathfrak {sl} _3 $的亚历山大多项式的非亚伯概括
A Non-Abelian Generalization of the Alexander Polynomial from Quantum $\mathfrak{sl}_3$
论文作者
论文摘要
亚历山大多项式的一种结构是与受限制的量子$ \ mathfrak {sl} _2 $在Unity的第四根根部相关的量子不变的。我们将这种结构概括为定义一个链接$δ_ {\ mathfrak {g}} $，对于任何半imple lie lie代数$ \ mathfrak $ n $的n $ n $ n $ n $ n $ n $ n $。专注于$ \ mathfrak {g} = \ mathfrak {sl} _3 $，我们建立了$δ_ {\ mathfrak {sl} _3} $与亚历山大·多项式之间的直接关系。我们表明，尽管$ r $ -matrix不满足亚历山大 - 孔道的关系，但$δ_{\ Mathfrak {\ Mathfrak {\ Mathfrak {\ Mathfrak {\ Mathfrak {\ mathfrak {s sl} _3} $的某些参数评估在这些点上都无法满足。我们将$δ_ {\ Mathfrak {sl} _3} $用于所有结节，最多七个十字架和其他各种示例，包括Kinoshita-terasaka结和Conway结突变对，这些突变对与此不变所区别。
One construction of the Alexander polynomial is as a quantum invariant associated with representations of restricted quantum $\mathfrak{sl}_2$ at a fourth root of unity. We generalize this construction to define a link invariant $Δ_{\mathfrak{g}}$ for any semisimple Lie algebra $\mathfrak{g}$ of rank $n$, taking values in $n$-variable Laurent polynomials. Focusing on the case $\mathfrak{g}=\mathfrak{sl}_3$, we establish a direct relation between $Δ_{\mathfrak{sl}_3}$ and the Alexander polynomial. We show that certain parameter evaluations of $Δ_{\mathfrak{sl}_3}$ recover the Alexander polynomial on knots, despite the $R$-matrix not satisfying the Alexander-Conway skein relation at these points. We tabulate $Δ_{\mathfrak{sl}_3}$ for all knots up to seven crossings and various other examples, including the Kinoshita-Terasaka knot and Conway knot mutant pair which are distinguished by this invariant.
