Splitting the spectral flow and the Alexander matrix
Splitting the spectral flow and the Alexander matrix
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分裂谱流和亚历山大矩阵
DOI：
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发表时间：
1994
期刊：
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通讯作者：
Daniel Ruberman
中科院分区：
文献类型：
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作者：
P. Kirk;E. Klassen;Daniel Ruberman
本文讨论了一个计算自伴算子路D_i = *dA_i-dA_i * 的谱流的方法，其中A_i是沿沿着环面分裂的3-流形Z上的SU（2）联络，A_o和A_i是平坦的。最近的定理吉田[Y1，Y2]表明如何进行这一点时，Z是通过手术获得的结，在一定的非退化条件。假设有一个路径A，平坦的连接上的结补和空间的平坦连接模规范变换是一个光滑的l维品种附近的这条道路，吉田表明与一个明确的公式，谱流是由限制的道路边界环面。作为我们的主要结果的后果，我们表明，当路径A，有奇点，谱流是不确定的限制边界环面。我们给出明确的计算在W比较的频谱流的手术的怀特黑德双结的频谱流的相应手术的怀特黑德双的unknot。这些例子在纽结补集上有平坦连接的路径，它们对边界的限制是相同的，而它们的谱流不同。设Z = XwY，其中X是S3中纽结的补.设A0和A1是Z上的平坦联络，它们对X的限制是可约的。然后在Z ' = X ' u Y上存在对应的平坦连接A ~和At，其中X'是非结补（即，实心环面）。在~4中我们证明了Z '上A~到AI的谱流与Z上Ao到A~的谱流之差是一个经典的纽结签名，并且实际上等于纽结的亚历山大矩阵的谱流。将此定理应用于卫星结，
This paper is concerned with a procedure for computing the spectral flow of a path of self-adjoint operators of the form D, = *dA, -dA,*, where the At are SU(2) connections on a 3-manifold Z which is split along a torus, and A0 and A~ are fiat. Recent theorems of Yoshida [Y1, Y2] show how to carry this out when Z is obtained by surgery on a knot, under certain nondegeneracy conditions. Under the assumption that there is a path A, of flat connections on the knot complement and that the space of flat connections modulo gauge transformation is a smooth l-dimensional variety near this path, Yoshida shows with an explicit formula that the spectral flow is determined by the restriction of the path to the boundary torus. As a consequence of our main result we show that when the path A, has singularities, the spectral flow is not determined by its restriction to the boundary torus. We give explicit computations in w comparing the spectral flow on a surgery of a Whitehead double of a knot to the spectral flow on the corresponding surgery of the Whitehead double of the unknot. These examples have paths of flat connections on the knot complements whose restrictions to the boundary are the same, while their spectral flows differ. Suppose Z = X w Y, where X is the complement of a knot in S 3. Let A0 and Al be flat connections on Z whose restrictions to X are reducible. Then there are corresponding flat connections A ~ and At on Z ' = X ' u Y, where X' is the unknot complement (i.e., a solid torus). In ~4 we show that the difference between the spectral flow from A~ to AI on Z ' and the spectral flow from Ao to A~ on Z is a classical knot signature, and in fact is equal to the spectral flow of the Alexander matrix of the knot. Applying this theorem to satellite knots yields examples in