The braid index of reduced alternating links
The braid index of reduced alternating links
复制标题
减少交替链的编织指数
DOI：
--
复制
发表时间：
2017
影响因子：
0.8
通讯作者：
Pengyu Liu
中科院分区：
文献类型：
--
作者：
Y. Diao;Gábor Hetyei;Pengyu Liu
众所周知，交错链环的最小交叉数等于该链环的任意简化交错链环图中的交叉数。这个显著的结果是琼斯多项式的一个应用。在交替链接的编织指数的情况下，山田表明，塞弗特圈的最小数量超过所有定期投影的链接等于编织指数。因此，人们可以猜想，在一个减少交替图的塞弗特圈的数量等于辫子指数的链接，但这原来是错误的。在本文中，我们证明了下一个最好的事情，人们可以希望：我们精确地搜索那些交替链接，其辫子指数等于塞弗特圈在其相应的减少交替链接图的数量。更具体地说，我们证明了，如果D是一个约化的交替链接图L，那么B（L），L的辫子指数，等于D中的塞弗特圈的数量当且仅当GS（D）不包含重量为1的边。这里GS（D），称为D的塞弗特图，是从D得到的边加权简单图，通过将D的每个塞弗特圈标识为GS（D）的顶点，使得GS（D）中的两个顶点由边连接当且仅当两个相应的塞弗特圈在D中共享它们之间的交叉，并且边的权重是两个塞弗特圈之间的交叉数。这一结果部分基于著名的MFW不等式，它指出L的HOMFLY多项式的a-跨度是2 B（L）−2的下界，以及山田将L的所有链图上的塞弗特圆的最小数目与B（L）联系起来的结果。
Abstract It is well known that the minimum crossing number of an alternating link equals the number of crossings in any reduced alternating link diagram of the link. This remarkable result is an application of the Jones polynomial. In the case of the braid index of an alternating link, Yamada showed that the minimum number of Seifert circles over all regular projections of a link equals the braid index. Thus one may conjecture that the number of Seifert circles in a reduced alternating diagram of the link equals the braid index of the link, but this turns out to be false. In this paper we prove the next best thing that one could hope for: we characterise exactly those alternating links for which their braid indices equal the numbers of Seifert circles in their corresponding reduced alternating link diagrams. More specifically, we prove that if D is a reduced alternating link diagram of an alternating link L, then b(L), the braid index of L, equals the number of Seifert circles in D if and only if GS(D) contains no edges of weight one. Here GS(D), called the Seifert graph of D, is an edge weighted simple graph obtained from D by identifying each Seifert circle of D as a vertex of GS(D) such that two vertices in GS(D) are connected by an edge if and only if the two corresponding Seifert circles share crossings between them in D and that the weight of the edge is the number of crossings between the two Seifert circles. This result is partly based on the well-known MFW inequality, which states that the a-span of the HOMFLY polynomial of L is a lower bound of 2b(L)−2, as well as the result of Yamada relating the minimum number of Seifert circles over all link diagrams of L to b(L).