Incompressibility of closed surfaces in toroidally alternating link complements
Incompressibility of closed surfaces in toroidally alternating link complements
复制标题
环形交替连杆补件中闭合表面的不可压缩性
DOI：
10.18910/3955
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发表时间：
1998
影响因子：
0.4
通讯作者：
K. Shimokawa
中科院分区：
文献类型：
--
作者：
K. Shimokawa
众所周知，交替的结和链接具有许多良好的特性，并且交替链接的概念已经有许多推广。例如，同质、充分、增强交替和几乎交替链接。 C.C. Adams 在 [1] 中引入了环形交替链接的概念。环形交替连杆类别包含交替连杆、几乎交替连杆、椒盐卷饼连杆和 Montesinos 连杆（参见 [1] 和 [3]）。 C. Hayashi 还在 [3] 中研究了任意正属表面上的交替链接。本文的目的是确定环形交替连杆补体中给定封闭可定向表面的不可压缩性。令 M 为透镜空间或 3 球体 S，令 T 为环面，它给出 M 的 Heegaard 分裂。该环面在同位素方面是唯一的（参见 [2]）。 M 中的连线 L 被称为环形交替（相对于 T），如果它可以同位素到 T 的邻域 T x / 中，使得它在 T 上具有交替图 ττ(L)，使得 T — ττ(L) 由相对于投影 π 的开圆盘组成：T x I —> T。令 F C M — L 为嵌入连接的闭合曲面。如果 F 是 2-球体并且 F 不限制 M-L 中的 3-球，或者 F 不是 2-球体并且对于每个圆盘 D C M — L 且 D Π F = dD，则称 F 在 M — L 中不可压缩，存在一个圆盘 D' C. F 且 θD = 3D'。如果不存在与 L 横向相交且 D Π F = 3D 的圆盘 D C M ，则 F 称为成对不可压缩。如果 F 的每个分量在 M — L 中不可压缩（或成对不可压缩），则 M L 中嵌入的不连续曲面 F 称为 M L 中不可压缩（或成对不可压缩）。如果嵌入 M L 中的每个 2-球体都限制了 M L 中的 3-球，则链接 L C M 被称为非分裂。W. Menasco 定义了无边界或具有交替结和链接补体中的子午边界的表面的标准位置概念
It has been known that alternating knots and links have many nice properties, and there have been many generalizations of the notion of alternating links. For instance, homogeneous, adequate, augmented alternating and almost alternating links. C.C. Adams introduced the notion of toroidally alternating links in [1]. The class of toroidally alternating links turned out to contain those of alternating links, almost alternating links, pretzel links and Montesinos links (See [1] and [3]). C. Hayashi also studied alternating links on surfaces of arbitrary positive genera in [3]. The purpose of this paper is to determine incompressibility of given closed orientable surfaces in toroidally alternating link complements. Let M be a lens space or the 3-sphere S, and let T be-a torus which gives a Heegaard splitting of M. This torus is unique up to isotopy (See [2]). A link L in M is called toroidally alternating (with respect to T) if it can be isotoped into a neighborhood T x / of T so that it has an alternating diagram ττ(L) on T such that T — ττ(L) consists of open discs with respect to a projection π : T x I —> T. Let F C M — L be an embedded connected closed surface. F is called incompressible in M — L if either F is a 2-sphere and F does not bound a 3-ball in M — L, or F is not a 2-sphere and for each disc D C M — L with D Π F = dD, there is a disc D' C. F with ΘD = 3D'. F is called pairwise incompressible if there does not exist a disc D C M meeting L transversely in one point with D Π F = 3D. An embedded disconnected surface F in M L is called incompressible in M L (resp. pairwise incompressible) if every component of F is incompressible in M — L (resp. pairwise incompressible). A link L C M is called non-split if every 2-sphere embedded in M L bounds a 3-ball in M L. W. Menasco defined the notion of standard position for surfaces either without boundary or with meridional boundary in alternating knot and link complements