一个减号，过去常常让我烦恼，但现在我知道它为什么在那里了
A minus sign that used to annoy me but now I know why it is there
摘要：
我们考虑两个著名的链接不变量的建设。其中一个使用绞合理论：你把链接的每一个交叉点都分解成不交叉的东西的线性组合，直到你最终得到一个没有交叉点的链接的线性组合，你把它变成一个多项式。另一种是使用量子群：你从一个拓扑范畴构造一个函子到某个表示范畴，这样（有向框架）链接就被发送到平凡表示的自同态，而平凡表示只是有理函数。这两种结构的某些实例产生了本质上相同的不变量，但是当人们仔细地将它们匹配时，会出现一个似乎不合适的减号。我们将讨论在琼斯多项式的情况下，这些结构是如何匹配的，以及减号是从哪里来的。在量子群方面，人们被引导使用一个非标准的带状元素，这使得人们可以考虑一个更大的拓扑范畴。
英文摘要：
We consider two well known constructions of link invariants. One uses skein theory: you resolve each crossing of the link as a linear combination of things that don't cross, until you eventually get a linear combination of links with no crossings, which you turn into a polynomial. The other uses quantum groups: you construct a functor from a topological category to some category of representations in such a way that (directed framed) links get sent to endomorphisms of the trivial representation, which are just rational functions. Certain instances of these two constructions give rise to essentially the same invariants, but when one carefully matches them there is a minus sign that seems out of place. We discuss exactly how the constructions match up in the case of the Jones polynomial, and where the minus sign comes from. On the quantum group side, one is led to use a non-standard ribbon element, which then allows one to consider a larger topological category.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
