Khovanov-Rozansky同源性和布线
Khovanov-Rozansky Homologies and Cabling
摘要：
在HOMFLY多项式的成缆过程中，一个结的有色HOMFLY多项式是由普通的HOMFLY加上额外的捻数得到的。因此，有色多项式可以看作是不同捻数的索结的HOMFLYs之间的关系。在目前的工作中，我们寻找这种类型的Khovanov-Rozansky同调的关系和调查，为什么没有推广Khovanov-Rozansky同调非反对称表示已建成。我们考虑最简单的可能情况，即在（11）-和（2）-表示中着色的非结。HOMFLY关系的朴素t-变形在这两种情况下都不存在。在（11）-表示的情况下，我们成功地切换到一个同调描述，并导致我们的猜想Khovanov-Rozansky同源环面结$T(2,k)$。（2）-情况只提供框架依赖的答案，没有简单的变换规则，如q，t-移位。我们已经表明，在我们的过程中，其非平凡的组件与不同的t-分级的数量取决于选择unknot框架。
英文摘要：
In the cabling procedure for HOMFLY polynomials colored HOMFLY polynomials of a knot are obtained from ordinary HOMFLY of the cabled knot with extra twists added. Thus colored polynomials can be seen as relation between HOMFLYs of cabled knot with different twists. In present work we search for relations of such type in Khovanov-Rozansky homologies and investigate, why no generalizations of Khovanov-Rozansky homologies to non-skew-symmetric representations have been constructed. We consider the simplest possible case, i.e. the unknot colored in (11)- and (2)-representations. Naive t-deformation of HOMFLY relation failed to exist in both cases. In case of (11)-representation we have succeeded after a switch to a homological description and that led us to a conjecture about Khovanov-Rozansky homologies of torus knots $T(2,k)$. The (2)-case provides only framing-dependent answers and no simple rule of transformation, such as q,t-shift is seen. We have shown that in our procedure the number of whose nontrivial components with different t-gradings depends on the choice of unknot framing.
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。
