Knot signature functions are independent
Knot signature functions are independent
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结签名函数是独立的
DOI：
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发表时间：
2002
期刊：
影响因子：
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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：
Jae Choon Cha;C. Livingston
塞弗特矩阵是满足det（V-VT）=±1的平方积分矩阵V。对于这样的矩阵和单位复数ω，存在对应于签名σ ω（V）= sign（（1 - ω）V +（1 - ω）VT）的签名。设S表示虚部为正的单位复数的集合。我们证明了{σ ω } ω ∈ S是线性无关的，可以看作是所有Seifert矩阵集合上的一组函数.如果V是代谢的，则σ ω（V）= 0，除非ω是亚历山大多项式的根，Δ V（t）= det（V-tVT）。设A表示所有虚部为正的亚历山大多项式的单位根的集合。我们证明了当把{σ ω } ω ∈ A看作是所有代谢Seifert矩阵集合上的一组函数时，它是线性无关的.对于每一个纽结KCS 3，可以关联一个塞弗特矩阵VK，并且σ ω（VK）导出一个纽结不变量。我们的结果的拓扑应用包括一个证明，函数的集合{σ ω } ω ∈ S是线性无关的集合上的所有节点和双边平均签名函数的集合，{σ* ω } ω ∈ S，形成一个线性无关的集合上的节点和谐群的同态。另外，如果v ∈ S是某个亚历山大多项式的根，则存在一个切片纽结K，其签名函数σ ω（K）仅在ω = v和ω = v处是非平凡的.
A Seifert matrix is a square integral matrix V satisfying det(V-V T )=±1. To such a matrix and unit complex number ω there corresponds a signature, σ ω (V) = sign((1 - ω)V + (1 - ω)V T ). Let S denote the set of unit complex numbers with positive imaginary part. We show that {σ ω } ω ∈ S is linearly independent, viewed as a set of functions on the set of all Seifert matrices. If V is metabolic, then σ ω (V) = 0 unless ω is a root of the Alexander polynomial, Δ V (t) = det(V - tV T ). Let A denote the set of all unit roots of all Alexander polynomials with positive imaginary part. We show that {σ ω } ω ∈ A is linearly independent when viewed as a set of functions on the set of all metabolic Seifert matrices. To each knot K C S 3 one can associate a Seifert matrix V K , and σ ω (V K ) induces a knot invariant. Topological applications of our results include a proof that the set of functions {σ ω } ω ∈ S is linearly independent on the set of all knots and that the set of two-sided averaged signature functions, {σ* ω } ω ∈ S , forms a linearly independent set of homomorphisms on the knot concordance group. Also, if v ∈ S is the root of some Alexander polynomial, then there is a slice knot K whose signature function σ ω (K) is nontrivial only at ω = v and ω = v. We demonstrate that the results extend to the higher-dimensional setting.