Concordance and mutation
Concordance and mutation
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一致性和变异
DOI：
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发表时间：
1999
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通讯作者：
C. Livingston
中科院分区：
文献类型：
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作者：
P. Kirk;C. Livingston
我们提供了一个框架，研究一致性和正突变之间的相互作用，并确定了一些有关的基本结构。
理解纽结协调的基本结果是Levine证明的结构定理：当n>1时，存在从S^{2n+1}中的纽结（2n-1）-球面的协调群Cn到代数定义的群G_{+-}的同构φ，并且G_{+-}同构于无限直和Z_2_infty直和Z_4_infty。这是一个令人吃惊的后果的工作卡森和戈登，在经典的情况下，核心的φ对C_1是无限产生的。除此之外，关于（C_1，phi）对的研究还很少。
本文提出了一种研究C_1的新方法，即引入群M，M定义为纽结集通过协调和正突变产生的等价关系的商，群运算由连通和导出。我们证明了存在φ的因子分解C_1-->M-->G_-。我们的主要结果是，这两个地图有无限生成的内核。
在经典纽结的几何构造中，最微妙的是正突变。正突变体是不可区分的使用经典的阿贝尔结不变量，以及由这样的现代不变量的琼斯，Homfly或考夫曼多项式。区分阳性突变体直到一致性是一个困难得多的问题;到目前为止只知道一个例子。在本文中的结果提供，除其他结果外，第一无限家庭的结是不同的，从他们的积极的突变体，甚至一致。
We provide a framework for studying the interplay between concordance and positive mutation and identify some of the basic structures relating the two.
The fundamental result in understanding knot concordance is the structure theorem proved by Levine: for n>1 there is an isomorphism phi from the concordance group C_n of knotted (2n-1)-spheres in S^{2n+1} to an algebraically defined group G_{+-}; furthermore, G__{+-} is isomorphic to the infinite direct sum Z^infty direct sum Z_2^infty direct sum Z_4^infty. It was a startling consequence of the work of Casson and Gordon that in the classical case the kernel of phi on C_1 is infinitely generated. Beyond this, little has been discovered about the pair (C_1,phi).
In this paper we present a new approach to studying C_1 by introducing a group, M, defined as the quotient of the set of knots by the equivalence relation generated by concordance and positive mutation, with group operation induced by connected sum. We prove there is a factorization of phi, C_1-->M-->G_-. Our main result is that both maps have infinitely generated kernels.
Among geometric constructions on classical knots, the most subtle is positive mutation. Positive mutants are indistinguishable using classical abelian knot invariants as well as by such modern invariants as the Jones, Homfly or Kauffman polynomials. Distinguishing positive mutants up to concordance is a far more difficult problem; only one example has been known until now. The results in this paper provide, among other results, the first infinite families of knots that are distinct from their positive mutants, even up to concordance.