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Primary decomposition of knot concordance and von Neumann rho-invariants
Proceedings of the American Mathematical Society ( IF 0.8 ) Pub Date : 2020-10-20 , DOI: 10.1090/proc/15282
Min Hoon Kim , Se-Goo Kim , Taehee Kim

Abstract:We address the primary decomposition of the knot concordance group in terms of the solvable filtration and higher order von Neumann $ \rho $-invariants by Cochran, Orr, and Teichner. We show that for a non-negative integer $ n$, if the connected sum of two $ n$-solvable knots with coprime Alexander polynomials is slice, then each of the knots has vanishing von Neumann $ \rho $-invariants of order $ n$. This gives positive evidence for the conjecture that nonslice knots with coprime Alexander polynomials are not concordant. As an application, we show that if $ K$ is one of Cochran-Orr-Teichner's knots which are the first examples of nonslice knots with vanishing Casson-Gordon invariants, then $ K$ is not concordant to any knot with Alexander polynomial coprime to that of $ K$.


中文翻译:

结一致性和冯·诺依曼变容常数的初次分解

摘要:我们$ \ rho $通过Cochran,Orr和Teichner的可解过滤和高阶冯·诺伊曼不变量解决了结一致基团的初次分解。我们证明,对于一个非负整数$ n $,如果两个可$ n $解结与互质素亚历山大多项式的连接和为切片,则每个结$ \ rho $的阶数为von Neumann-不变式$ n $。这为猜想提供了积极的证据,即非互结与共质数亚历山大多项式不一致。作为一个应用,我们表明,如果$ K $是Cochran-Orr-Teichner的结中的一个,这是具有Casson-Gordon不变量消失的非切片结的第一个示例,那么$ K $与亚历山大多项式互质数的任何结都不相符。$ K $
更新日期:2020-11-12
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