Mutation and volumes of knots inS3
Mutation and volumes of knots inS3
复制标题
S3 中结的突变和体积
DOI：
10.1007/bf01389038
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发表时间：
1987
影响因子：
3.1
通讯作者：
Daniel Ruberman
中科院分区：
文献类型：
--
作者：
Daniel Ruberman
《S 3》中的节点可以用几种不同的方式分解成更简单的片段。最基本的是把和连成质数；这样的分解是唯一的。此外，很容易理解每个被和项对希望计算的任何纽结不变项的贡献。另一种分裂是当S 3中的一个2-球(称为康威球)在四个点上横向打结时。由此产生的分裂成所谓的纠缠在各种对称性和纽结的其他性质的研究[7，8，22，6]中被证明是相当有成效的。给定一个康威球体，有一种称为突变的操作，它会产生一个新的结。粗略地说，一个人拿出缠绕在一起的东西，把它翻过来，然后粘回去。结果结往往与原始结不同，除非一侧的缠结是对称的[6]。然而，许多纽结的不变量通过突变来保持[7]，例如签名和Alexander多项式，以及新的二元纽结多项式[12]。本文证明了纽结及其突变体的Gromov范数(见下文)重合。特别地，如果S3-K是双曲流形，那么K的S3-突变体也是双曲流形，并且它们的体积是相同的。这些结果是一个更一般的定理的例子，该定理表明格罗莫夫范数通过沿曲面的某些剪切和粘贴保持不变。通过使用一般三维流形的环面分解，我们可以将问题归结为理解双曲流形会发生什么。对于双曲流形，我们证明了对于某些曲面FcM和F的对称性r，沿F切割M并通过z调整得到一个新的双曲流形M‘，其中Vol(M)-Vol(M’)。Colin Adams[1]对M中三次穿孔球面的特例证明了类似的结果。我们的基本方法可以推广到，对于产生双曲流形的纽结上的Dehn手术，其体积与在突变纽结上的相应手术相同。我们还给出了关于纽结及其突变体的分枝循环覆盖体积的类似结果。在以后的文章中，我们将考虑这种剪切和粘贴对双曲流形的Chern-Simons不变量和~/-不变量的影响，在证明关于双曲流形的定理的过程中，我们需要使用[1]关于最小面积曲面的嵌入性和交集的结果。文[11]中的定理只涉及紧致情形，而在有限体积双曲三维流形的情形下，我们需要它们的类似结果。这些
Knots in S 3 can be decomposed into simpler pieces in several different ways. The most basic is by connected sum into prime pieces; such a decomposition is unique. Further, it is easy to understand the contribution of each summand to any knot invariant one might wish to compute. Another splitting of knots occurs when a 2-sphere (called a Conway sphere) in S 3 hits the knot transversally in four points. The resulting splitting into so-called tangles has proved quite fruitful in various investigations [7, 8, 22, 6] of symmetries and other properties of knots. Given a Conway sphere, there is an operation called mutation which yields a new knot. Roughly speaking, one takes out the tangle, flips it over and glues it back in. The resulting knot tends to differ from the original one, unless the tangle on one side was symmetric [6]. However, many invariants of a knot are preserved by mutation [7], eg the signature and Alexander polynomial, as well as the new two-variable knot polynomial [12]. In this paper we show that the Gromov norm (see below) of a knot and its mutant coincide. In particular if S3--K is a hyperbolic manifold then S3-mutant of K is as well, and their volumes are the same. These results are instances of a more general theorem which shows that Gromov's norm is preserved by certain kinds of cutting and pasting along surfaces. By using the torus decomposition of a general 3-manifold, one reduces the problem to understanding what happens for hyperbolic manifolds. For the case of a hyperbolic manifold we show that for certain surfaces F c M and symmetries r of F, cutting M along F and regluing via z results in a new hyperbolic manifold M'with vol (M)--vol (M'). Colin Adams [1] has proved a similar result for the special case of a thrice-punctured sphere in M. Our basic method can be extended to show that for Dehn surgeries on a knot which produce hyperbolic manifolds, the volume is the same for the corresponding surgeries on the mutant knot. We also give a similar result about the volumes of the branched cyclic covers of the knot and its mutant. In a future paper, we will consider the effect of this sort of cutting and pasting on the Chern-Simons invariant and~/-invariant of hyperbolic manifolds.In the course of the proof of the theorem concerning hyperbolic manifolds, we need to use the results of [I1] on embeddedness and intersections of least area surfaces. The theorems of [11] deal only with the compact case, and we need their analogues in the case of finite-volume hyperbolic 3-manifolds. These
一组穿孔曲面SL(2,C)表示空间的坐标系及其应用
DOI：
--
发表时间：
2008
期刊：
影响因子：
--
作者：
S.Albeverio;S.Liang;S.Liang;T.Nakanishi;中西敏浩;名和範人;服部哲弥;服部哲弥;名和範人;服部久美子;中西敏浩
通讯作者：
中西敏浩