Surface-Knots in 4-Space
Surface-Knots in 4-Space
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4 空间中的表面结
DOI：
10.1007/978-981-10-4091-7
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发表时间：
2017
影响因子：
0.6
通讯作者：
S. Kamada
中科院分区：
文献类型：
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作者：
S. Kamada
纽结理论是现代数学中最活跃的研究领域之一。纽结和链环是欧几里得三维空间中的闭曲线（一维流形），它们与辫子和三维流形有关。这些概念被推广到更高的维度。曲面纽结和曲面链环是欧氏四维空间中的闭曲面（二维流形），它们与二维辫和四维流形有关。曲面纽结理论不仅处理闭曲面，而且处理四维流形中的带边界曲面。例如纽结协调和纽结配边，它们也是纽结理论中的重要对象，它们是3-球面和区间的乘积空间中的曲面。Artin在20世纪20年代，RH Fox使用电影方法开始了关键的研究，在20世纪60年代，J. Milnor随后进行了包括结一致性研究在内的研究。在三维空间中使用曲面图的研究始于D.罗斯曼在20世纪70年代，并已广泛做了JS卡特和M。自1990年代以来，齐藤。作者从20世纪90年代开始研究二维编织物的表面结。建立了类似于亚历山大定理和马尔可夫定理的面结编织定理。自20世纪90年代后期以来，人们利用Quandles及其（上）同调理论研究了纽结和曲面纽结的不变量。JS Carter，D. Jelsovsky，S.卡马达湖Langford和M.斋藤（CJKLS）构建的不变量称为quandle上循环不变量，他们的不变量现在被扩展和推广到各种不变量，使他们被用来研究手征的结，双曲卷和陈和西蒙的不变量，可逆的表面结，三重点的表面结数，等这本书的组织如下：第1章是专门介绍和pasteraries。在Chap. 2、介绍了纽结理论的基本知识。在Chap. 3、介绍了运动图像法和用带标记的经典图描述曲面结点的方法。如何计算一个表面结从运动图片的结组也解释了那里。图在3-空间的表面结和不变量得到的图表在第章。4.我们将在第二章讨论1-句柄与曲面结点的连接。5.第二章介绍了2-纽结和纽结射影平面的旋转构造。6.结一致性和结
Knot theory is one of the most active research fields in modern mathematics. Knots and links are closed curves (1-dimensional manifolds) in the Euclidean 3-space, and they are related to braids and 3-manifolds. These notions are generalized into higher dimensions. Surface-knots and surface-links are closed surfaces (2-dimensional manifolds) in the Euclidean 4-space, and they are related to 2-dimensional braids and 4-manifolds. Surface-knot theory treats not only closed surfaces but also surfaces with boundaries in 4-manifolds. For example, knot concordance and knot cobordism, that are also important objects in knot theory, are surfaces in the product space of the 3-sphere and the interval.Although the beginning of the study of surface-knots is due to E. Artin in the 1920s, the crucial research was started by RH Fox using the motion picture method, in the 1960s, followed by J. Milnor including researches on knot concordance. Studies using surface diagrams in 3-space were started by D. Roseman in the 1970s and have been extensively done by JS Carter and M. Saito since the 1990s. The author has been studying surface-knots using 2-dimensional braids since the 1990s. Theorems on braiding of surface-knots analogous to Alexander and Markov’s theorems have been established. Since the late 1990s, invariants of knots and surface-knots using quandles and their (co-) homology theory have been studied. JS Carter, D. Jelsovsky, S. Kamada, L. Langford, and M. Saito (CJKLS) constructed invariants called the quandle cocycle invariants, and their invariants are now extended and generalized to various invariants so that they are used to study chirality of knots, hyperbolic volumes and Chern and Simon’s invariant, invertibility of surface-knots, triple point numbers of surface-knots, etc. This book is organized as follows: Chapter 1 is devoted to an introduction and preliminaries. In Chap. 2, we introduce basics of knot theory. In Chap. 3, the motion picture method and a method describing surface-knots by classical diagrams with markers are introduced. How to compute the knot group of a surface-knot from a motion picture is also explained there. Diagrams in 3-space of surface-knots and invariants obtained from diagrams are treated in Chap. 4. We discuss 1-handles attaching to surface-knots in Chap. 5. Spinning constructions of 2-knots and knotted projective planes are introduced in Chap. 6. Knot concordance and knot