符号除子的Khovanov同调
Khovanov homology for signed divides
摘要：
利用Khovanov同调论解释强可逆链环的多项式不变量。一个划分，这是一个适当的一般浸入的有限数量的副本的单位间隔和圆的2-光盘，人们可以联想到一个强可逆链接的3-球。这可以推广到有符号除法：对每个交叉点分配+或-符号的除法。相反，对于任何对合$j$强可逆的链$L$，可以关联一个有符号除。两个通过关于对合的合痕而同位素的强可逆链称为强等价的。这种同位素会引起分歧。在作者以前的一篇论文中，可以找到一个详尽的清单，保持强等价的移动，连同这些移动的多项式不变，因此给出了相关强可逆链接的强等价不变。本文证明了该多项式可看作向量空间分次复形的分次欧拉示性数。这类复形的同调对于整除上的移动是不变的，因此通过强可逆链的强等价是不变的。
英文摘要：
The purpose of this paper is to interpret polynomial invariants of strongly invertible links in terms of Khovanov homology theory. To a divide, that is a proper generic immersion of a finite number of copies of the unit interval and circles in a 2-disc, one can associate a strongly invertible link in the 3-sphere. This can be generalized to signed divides : divides with + or - sign assignment to each crossing point. Conversely, to any link $L$ that is strongly invertible for an involution $j$, one can associate a signed divide. Two strongly invertible links that are isotopic through an isotopy respecting the involution are called strongly equivalent. Such isotopies give rise to moves on divides. In a previous paper of the author, one can find an exhaustive list of moves that preserves strong equivalence, together with a polynomial invariant for these moves, giving therefore an invariant for strong equivalence of the associated strongly invertible links. We prove in this paper that this polynomial can be seen as the graded Euler characteristic of a graded complex of vector spaces. Homology of such complexes is invariant for the moves on divides and so is invariant through strong equivalence of strongly invertible links.
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