We introduce a numerical invariant called the braid alternation number that measures how far a link is from being an alternating closed braid. This invariant resembles the alternation number, which was previously introduced by the second author. However, these invariants are not equal, even for alternating links. We study the relation of this invariant with others and calculate this invariant for some infinite knot families. In particular, we show arbitrarily large gaps between the braid alternation number and the alternation and unknotting numbers. Furthermore, we estimate the braid alternation number for prime knots with nine crossings or less.

中文翻译：

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在交替闭合的辫子上

我们引入了一个称为编织交替数的数值不变量，用于测量链接距离交替闭合编织的距离。这个不变量类似于第二作者之前介绍的交替数。然而，这些不变量并不相等，即使对于交替链接也是如此。我们研究了这个不变量与其他不变量的关系，并为一些无限的结族计算了这个不变量。特别是，我们显示了编织交替数与交替和未打结数之间的任意大差距。此外，我们估计了具有九个或更少交叉的素结的编织交替数。