We characterize the fractional Dehn twist coefficient of a braid in terms of a slope of the homogenization of the Upsilon function, where Upsilon is the function‐valued concordance homomorphism defined by Ozsváth, Stipsicz, and Szabó. We use this characterization to prove that $n$‐braids with fractional Dehn twist coefficient larger than $n-1$ realize the braid index of their closure. As a consequence, we are able to prove a conjecture of Malyutin and Netsvetaev stating that $n$‐times twisted braids realize the braid index of their closure. We provide examples that address the optimality of our results. The paper ends with an appendix about the homogenization of knot concordance homomorphisms.

中文翻译：

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股线完全缠绕的辫子可实现辫子指数

我们用Upsilon函数均质化的斜率来表征辫子的分数Dehn扭曲系数，其中Upsilon是由Ozsváth，Stipsicz和Szabó定义的函数值一致同态。我们用这个特征来证明$ñ$-分数Dehn扭曲系数大于的编织物 $ñ-\mathrm{1个}$实现其关闭的编织索引。结果，我们能够证明马留丁和内茨维塔耶夫的猜想，$ñ$时代的扭曲辫子实现了其闭合的辫子指数。我们提供了可以解决我们的结果最优性的例子。本文以关于结一致性同态同构的同质化的附录结尾。