Polynomial One-cocycles for Knots and Closed Braids
Polynomial One-cocycles for Knots and Closed Braids
复制标题
结和闭合辫子的多项式一循环
DOI：
10.1142/11551
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
T. Fiedler
中科院分区：
文献类型：
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作者：
T. Fiedler
本书中的不变量源于主题的一个细微变化：我们研究单参数结族而不是单个结。这在数学中并不罕见：对于一个困难的问题，有时攻克一个更困难的问题是有用的（例如，在所有结图的模空间中寻找循环的不变量）。有向3流形中结的拓扑模空间是所有光滑结与给定结光滑同位素的无限维空间。如果3-流形是表面上的线束，那么我们考虑沿线到表面上的投影pr。在本书中，我们将假设线束是平凡的，因此曲面是可定向的（例如pr: p3 \{P t}→2仍然是一个有趣的进一步研究的例子）。此外，我们确定了线的方向，从而确定了曲面的方向。模空间中的每个点现在结图对公关,我们称这种额外的模空间结构的模空间结图,用M M的连接组件对应于一个结型K有时用M K .公关投影引起的地层分层在M .余维数0是普通结图及其补充M是奇异结的判别图,用Σ。M中的一个有向的一般环与Σ横相交于有限个数的点，这些点都属于M中co维数为1的地层Σ(1)。这些地层与Reidemeister运动完全对应，参见eg[12]。我们所有的图解1-环都是以下形式：我们将一个整数关联到s∩Σ(1)中的每个点，即Reidemeister移动，我们不关联任何与普通图…
The invariants in this book have their origin in a slight change of the subject: we study 1-parameter families of knots instead of individual knots. This is not an unusual feature in mathematics: for a difficult problem it is sometimes useful to attack an even more difficult problem (eg finding invariants for loops in the moduli space of all diagrams of knots). The topological moduli space of a knot in an oriented 3-manifold is the infinite dimensional space of all smooth knots smoothly isotopic to the given knot. If the 3-manifold is a line bundle over a surface then we consider the projection pr along the lines onto the surface. In this book we will assume that the line bundle is trivial and hence the surface is orientable (eg pr: ℝ P 3\{p t}→ ℝ P 2 stays an interesting case for further investigations). Moreover, we fix an orientation of the lines and hence of the surface. Each point in the moduli space is now a knot diagram with respect to pr and we call the moduli space with this additional structure the moduli space of knot diagrams, denoted by M. The connected component of M which corresponds to a knot type K is sometimes denoted by M K. The projection pr induces a stratification on M. The strata of codimension 0 are the ordinary knot diagrams and their complement in M is the discriminant of singular knot diagrams, denoted by Σ. An oriented generic loop in M intersects Σ transversely in a finite number of points, which belong all to strata Σ (1) of codimension 1 in M. These strata correspond exactly to the Reidemeister moves, see eg [12]. All our diagrammatic 1-cocycles are of the following form: we associate an integer to each point in s∩ Σ (1), ie the Reidemeister moves, and we associate nothing to the ordinary diagrams…