Categorification of the Kauffman bracket skein module of I-bundles over surfaces
Categorification of the Kauffman bracket skein module of I-bundles over surfaces
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表面工字束考夫曼支架绞纱模块的分类
DOI：
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发表时间：
2004
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通讯作者：
Adam S. Sikora
中科院分区：
文献类型：
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作者：
M. Asaeda;J. Przytycki;Adam S. Sikora
Khovanov定义了环LR 3的分次同调群，并证明了它们的多项式Euler特征线是L的Jones多项式. Khovanov的构造不能直接推广到曲面F6 D2上的I-丛M中的链（只有Z/2系数的同调除外）。因此，本文的目标是提供一个非平凡的推广，他的方法导致同源不变量的链接在M与任意环的系数。在证明了我们的同调群在Reidemeister移动下的不变性之后，我们证明了L的同调群的多项式Euler特征线决定了L在M的skein模的标准基中的系数.因此，我们的同调群提供了M的Kauffman括号串模的一个“自同构”.此外，我们证明了我们的同调群的Viro的确切序列的推广。最后，我们证明了一个对偶定理，它将任意环L的上同调群与L的镜像的同调群联系起来。AMS分类57 M27; 57 M25，57 R56
Khovanov defined graded homology groups for links LR 3 and showed that their polynomial Euler characteristic is the Jones polyno- mial of L. Khovanov's construction does not extend in a straightforward way to links in I-bundles M over surfaces F 6 D 2 (except for the homol- ogy with Z/2 coefficients only). Hence, the goal of this paper is to provide a nontrivial generalization of his method leading to homology invariants of links in M with arbitrary rings of coefficients. After proving the invariance of our homology groups under Reidemeister moves, we show that the polynomial Euler characteristics of our homology groups of L determine the coefficients ofL in the standard basis of the skein module of M. Therefore, our homology groups provide a "categorifi- cation" of the Kauffman bracket skein module of M. Additionally, we prove a generalization of Viro's exact sequence for our homology groups. Finally, we show a duality theorem relating cohomology groups of any link L to the homology groups of the mirror image of L. AMS Classification 57M27; 57M25, 57R56