An algebraic classification of some links of codimension two
An algebraic classification of some links of codimension two
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DOI：
10.1090/s0002-9939-1977-0458439-1
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发表时间：
1977
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通讯作者：
C. Liang
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文献类型：
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作者：
C. Liang
对于 q > 2，J. Levine 证明了两个简单的 (2q 1)-结是同位素当且仅当它们的 Seifert 矩阵相等。在本文中，我们将证明简单边界 (2q 1) 链接的 Levine 结果的类比；我们将证明：“对于 q > 3，两个简单边界 (2q 1) 链接是同位素当且仅当它们的 Seifert 矩阵是 I 等价的（由一些代数移动定义）。”重数 m 的 n 链接，表示为 L = K1 U ... U Km 是将 n 球体（或同伦球体）Ki 的 m 个不相交副本嵌入到 (n + 2) 球体 Sn+2 中。如果 L 扩展到 m 个不相交的可定向紧致 (n + 1) 流形 Mi 的嵌入（称为 Seifert 流形），且 aMi = Ki，则称为边界。让 X 表示链接补集。 Gutierrez [1] 表明，当且仅当存在从 71(X) 到 Fm（m 个生成器中的自由群）的同态，将经线发送到生成器时，重数 m 的 n 链接才是边界。如果 i 2 的 7Ti(X) = 7Ti(VmS1)，则 (2q 1)-link L 被称为简单结，Levine [5] 证明了两个简单的 (2q 1)-结是同位素当且仅当它们的 Seifert 矩阵是“等价的”（由 [5] 中的某些代数“移动”定义，在 [7] 中也称为 S 等价）。在本文中，我们将证明简单边界 (2q 1) 链接的莱文定理 1-3 的类似物，q > 3：两个简单边界 (2q 1) 链接是同位素当且仅当它们的“Seifert 矩阵”通过某些代数“移动”相关时。由于我们的证明与[4]和[5]几乎相同，因此我们在这里仅给出概述。 1. 为简单起见，我们将仅考虑重数 2 的 (2q 1)-link。这里考虑的所有内容都属于平滑类别。令 L = K1 u K2 为边界 (2q 1) 链接。根据[1]，L存在两个不相交的2q维Seifert流形Ml和M2，即WI=K1和3M2=K2。令 A 1 为 1977 年 1 月 13 日编辑收到的和 1977 年 2 月 28 日修订后的 AMS (MOS) 主题分类 (1970) 的相应 Seifert 矩阵。主要 57C45、57D40、57D65。
For q > 2, J. Levine proved that two simple (2q 1)-knots are isotopic if and only if their Seifert matrices are equivalent. In this paper, we will prove the analogue of Levine's result for simple boundary (2q 1)links; we will show that: "For q > 3, two simple boundary (2q 1)-links are isotopic if and only if their Seifert matrices are I-equivalent (defined by some algebraic moves)." An n-link of multiplicity m, denoted by L = K1 U ... U Km is an embedding of m disjoint copies of the n-sphere (or homotopy spheres) Ki into the (n + 2)-sphere Sn+2. L is called boundary if it extends to an embedding of m disjoint orientable compact (n + 1)-manifolds Mi, called the Seifert manifolds, with aMi = Ki. Let X denote the link complement. Gutierrez [1] showed that an n-link of multiplicity m is boundary if and only if there is an epimorphism from 71(X) onto Fm, the free group in m generators, sending meridians to generators. An (2q 1)-link L is called simple if 7Ti(X) = 7Ti(VmS1) for i 2, Levine [5] proved that two simple (2q 1)-knots are isotopic if and only if their Seifert matrices are "equivalent" (defined by certain algebraic "moves" in [5], also called S-equivalent in [7]). In this paper, we will prove the analogue of Levine's Theorems 1-3 for simple boundary (2q 1)links, q > 3: two simple boundary (2q 1)-links are isotopic if and only if their "Seifert matrices" are related by certain algebraic "moves". Since our proofs are almost the same as those of [4] and [5], we will only give the outlines here. 1. For simplicity, we will consider only the (2q 1)-link of multiplicity 2. Everything considered here is in the smooth category. Let L = K1 u K2 be a boundary (2q 1)-link. According to [1], there exist two disjoint 2q-dimensional Seifert manifolds Ml and M2 for L, that is, WI= K1 and 3M2 = K2. Let A 1 be the corresponding Seifert matrix for the Received by the editors January 13, 1977 and, in revised form, February 28, 1977. AMS (MOS) subject classifications (1970). Primary 57C45, 57D40, 57D65.