论文标题
彩色琼斯和亚历山大多项式作为配置空间中循环的拓扑交集
Coloured Jones and Alexander polynomials as topological intersections of cycles in configuration spaces
论文作者
论文摘要
有色琼斯和亚历山大多项式是量子不变的序列，以第一项恢复了琼斯和亚历山大多项式。我们表明，使用拓扑工具，可以以相同的方式在概念上看到它们，作为覆盖拉格朗日submanifolds显式同源类别的空间中的相交配对。主要结果证明，$ n^{th} $彩色的琼斯多项式和$ n^{th} $彩色的亚历山大·多项式出现，因为在两个变量上相同同源类的交叉配对的不同专业化是不同的，在每种情况下都有额外的帧校正。第一个推论从量子的角度解释了Bigelow的琼斯多项式和叉子的图片。其次，我们得出结论，$ n^{th} $彩色亚历山大多项式是在$ \ mathbb z \ oplus \ oplus \ mathbb z_n $中划分的交点配对。
Coloured Jones and Alexander polynomials are sequences of quantum invariants recovering the Jones and Alexander polynomials at the first terms. We show that they can be seen conceptually in the same manner, using topological tools, as intersection pairings in covering spaces between explicit homology classes given by Lagrangian submanifolds. The main result proves that the $N^{th}$ coloured Jones polynomial and $N^{th}$ coloured Alexander polynomial come as different specialisations of an intersection pairing of the same homology classes over two variables, with extra framing corrections in each case. The first corollary explains Bigelow's picture for the Jones polynomial with noodles and forks from the quantum point of view. Secondly, we conclude that the $N^{th}$ coloured Alexander polynomial is a graded intersection pairing in a $ \mathbb Z \oplus \mathbb Z_N$-covering of the configuration space in the punctured disc.
