This paper concerns the braid index of an alternating link. It is well known that the braid index of any link equals the minimum number of Seifert circles among all link diagrams representing it. For an alternating link represented by a reduced alternating diagram D, it is known that s(D), the number of Seifert circles in D, equals the braid index b(D) of D if D contains no lone crossings, where a crossing in D is called a lone crossing if it is the only crossing between two Seifert circles in D. If D contains lone crossings, then one can reduce the number of Seifert circles in D using link-type preserving moves such as the Murasugi–Przytycki operation. Let r(D) ≥ 1 be the maximum number of Seifert circles in D that can be reduced, then we have b(D) ≤ s(D) − r(D). On the other hand, if the a-span of the HOMFLY polynomial of D satisfies the equality a-span/2 + 1 = s(D) − r(D), then the Morton–Frank–Williams (MFW) inequality a-span/2 + 1 ≤b(D) leads us to the simple braid index formula b(D) = s(D) − r(D). In this paper, we derive explicit formulas for many alternating links based on minimum projections of these links by establishing the equality a-span/2 + 1 = s(D) − r(D). Our methods depend on the structures of the link diagrams under consideration and our results lead to explicit braid index formulas that are applicable to a very large class of links, a proper subset of which contains all two bridge links, all alternating pretzel links, and more generally all alternating Montesinos links. The derived braid index formula for an alternating Montesinos link is a function whose inputs are the signs of the crossings in the rational tangles of the Montesinos link. Finally, by applying the now proven Jones Conjecture on the writhe of minimum braids of a link, our results also allow us to obtain explicit formulas of the writhe of minimum braids for the links discussed in this paper from the minimum projections of these links.

中文翻译：

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一种确定交替链接编织指数的图解方法

本文涉及交变链接的编织指数。众所周知，任何链路的编织指数等于所有表示它的链路图中的最小 Seifert 圆数。对于由简化交替图表示的交替链接D， 众所周知s(D), 中的 Seifert 圈数D, 等于编织指数b(D)的D如果D不包含孤独的过境点，其中有一个交叉口D被称为孤独的穿越如果它是两个 Seifert 圆之间的唯一交叉点D. 如果D包含孤立的交叉点，那么可以减少 Seifert 圈的数量D使用诸如 Murasugi–Przytycki 操作之类的链接类型保留动作。让r(D) ≥ 1是最大的 Seifert 圈数D可以减少，那么我们有b(D) ≤ s(D) - r(D). 另一方面，如果一种-HOMFLY 多项式的跨度D满足等式一种-跨度/2 + 1 = s(D) - r(D), 然后是 Morton-Frank-Williams (MFW) 不等式一种-跨度/2 + 1 ≤b(D)将我们引向简单的编织指数公式b(D) = s(D) - r(D). 在本文中，我们通过建立等式，基于这些链接的最小投影推导出许多交替链接的显式公式一种-跨度/2 + 1 = s(D) - r(D). 我们的方法取决于所考虑的链接图的结构，我们的结果导致明确的编织指数公式适用于非常大的链接类别，其中一个适当的子集包含所有两个桥链接，所有交替的椒盐脆饼链接等等通常所有交替的 Montesinos 链接。导出的交替 Montesinos 链接的编织指数公式是一个函数，其输入是 Montesinos 链接的有理缠结中的交叉符号。最后，通过将现已证明的琼斯猜想应用于链接的最小辫子的扭动，我们的结果还允许我们从这些链接的最小投影中获得本文讨论的链接的最小辫子的扭动的明确公式。