辫子共轭类的下确界、上确界和测地线长度
The infimum, supremum and geodesic length of a braid conjugacy class
摘要：
辫群B_n中共轭问题的数学解分别由Elrifai-Morton和作者于1994年和1998年给出。这两个解决方案产生两个共轭类不变量，被称为“inf”和“sup”。这两篇论文都没有解决的一个问题是一个人必须“循环”的次数m（分别是：“decycle”），以增加inf（相应地，减少sup）或确保它已经是最大的（分别地，最小的）对于给定的共轭类。我们的主要结果是证明了在第二种算法的情形下，m的上界为n-2，在第一种算法的情形下，上界为（（n^2-n）/2）-1。作为推论，我们证明了在这两种算法中，inf和sup的计算在字长和辫子指数上都是多项式的。整数inf和sup确定（但不是由）共轭类中元素的最短测地线长度，如Charney定义的，因此我们也得到了计算此测地线长度的多项式时间算法。
英文摘要：
Algorithmic solutions to the conjugacy problem in the braid groups B_n were given by Elrifai-Morton in 1994 and by the authors in 1998. Both solutions yield two conjugacy class invariants which are known as `inf' and `sup'. A problem which was left unsolved in both papers was the number m of times one must `cycle' (resp. `decycle') in order to increase inf (resp. decrease sup) or to be sure that it is already maximal (resp. minimal) for the given conjugacy class. Our main result is to prove that m is bounded above by n-2 in the situation of the second algorithm and by ((n^2-n)/2)-1 in the situation of the first. As a corollary, we show that the computation of inf and sup is polynomial in both word length and braid index, in both algorithms. The integers inf and sup determine (but are not determined by) the shortest geodesic length for elements in a conjugacy class, as defined by Charney, and so we also obtain a polynomial-time algorithm for computing this geodesic length.
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