SIAM Journal on Discrete Mathematics, Volume 34, Issue 3, Page 1944-1968, January 2020.
A rational link may be represented by any of the (infinitely) many link diagrams corresponding to various continued fraction expansions of the same rational number. The continued fraction expansion of the rational number in which all signs are the same is called a nonalternating form, and the diagram corresponding to it is a reduced alternating link diagram, which is minimum in terms of the number of crossings in the diagram. Famous formulas exist in the literature for the braid index of a rational link by Murasugi and for its HOMFLY polynomial by Lickorish and Millett, but these rely on a special continued fraction expansion of the rational number in which all partial denominators are even (called the all-even form). In this paper we present an algorithmic way to transform a continued fraction given in nonalternating form into the all-even form. Using this method we derive formulas for the braid index and the HOMFLY polynomial of a rational link in terms of its reduced alternating form, or, equivalently, the nonalternating form of the corresponding rational number.

中文翻译：

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简化交替图表示的有理链接的不变量

SIAM离散数学杂志，第34卷，第3期，第1944-1968页，2020年1月。
有理链接可以由（无限）许多与相同有理数的各个连续分数展开相对应的链接图表示。所有符号都相同的有理数的连续小数展开被称为非交替形式，并且与之对应的图是精简的交替链接图，这在图中的交叉次数方面最小。文献中存在着关于Murasugi的有理链的编织指数以及Lickorish和Millett的HOMFLY多项式的著名公式，但这些公式依赖于有理数的特殊连续分数展开式，其中所有部分分母都是偶数（称为全部-偶数形式）。在本文中，我们提出了一种将非交替形式给出的连续分数转换为全偶数形式的算法。