枕套与纽结群的无迹表示的扰动
The pillowcase and perturbations of traceless representations of knot groups
摘要：
我们引入了明确的完整扰动的Chern-Simons功能的3球包含一对unknotted弧。这些扰动给出了一个具体的局部方法，使得与Kronheimer和Mrowka的奇异瞬子结同调相关的平坦奇异SO（3）联络的模空间是非退化的.这项研究的机制是一个（拉格朗日）相交图，通过表示的限制，从一个结的缠结分解。当其中一个缠结是平凡的，我们的扰动使我们能够研究两个拉格朗日孤立的交叉点，以产生奇异瞬子结同源的最小生成集。这个相交处的（辛）流形对应于四穿孔2-球面的无迹特征簇，我们将其与熟悉的枕套相识别。我们调查的形象，在这个枕套的无痕表示的缠结通过删除一个平凡的纠缠2桥结和环面结。利用这一点，我们计算了各种环面结的奇异瞬子同调。在许多情况下，我们的计算使我们能够了解非平凡的差异，在谱序列从Khovanov同源奇异瞬子同源。
英文摘要：
We introduce explicit holonomy perturbations of the Chern-Simons functional on a 3-ball containing a pair of unknotted arcs. These perturbations give us a concrete local method for making the moduli spaces of flat singular SO(3) connections relevant to Kronheimer and Mrowka's singular instanton knot homology non-degenerate. The mechanism for this study is a (Lagrangian) intersection diagram which arises, through restriction of representations, from a tangle decomposition of a knot. When one of the tangles is trivial, our perturbations allow us to study isolated intersections of two Lagrangians to produce minimal generating sets for singular instanton knot homology. The (symplectic) manifold where this intersection occurs corresponds to the traceless character variety of the four-punctured 2-sphere, which we identify with the familiar pillowcase. We investigate the image in this pillowcase of the traceless representations of tangles obtained by removing a trivial tangle from 2-bridge knots and torus knots. Using this, we compute the singular instanton homology of a variety of torus knots. In many cases, our computations allow us to understand non-trivial differentials in the spectral sequence from Khovanov homology to singular instanton homology.
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