Toroidally alternating knots and links
Toroidally alternating knots and links
复制标题
环形交替的结和链节
DOI：
10.1016/0040-9383(94)90017-5
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发表时间：
1994
期刊：
影响因子：
--
通讯作者：
C. Adams
中科院分区：
文献类型：
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作者：
C. Adams
S3中的交错链环类有各种各样的推广，特别是交错链环、伪交错链环、齐次链环、适当链环、增广交错链环和几乎交错链环。这些概括起源于试图将已知的交替链接的结果扩展到更广泛的链接类别。本文将交错环类推广到一个新的集合，我们称之为环交错环。这一环链的集合将特别广泛，其中包含交替环链的集合、几乎交替环链的集合、具有单个增广分量的增广交替环链的子集以及树环链集合的子类，包括所有的蒙特西诺斯环链。令人惊讶的是，如果检查[6]中出现的通过11个交叉点的素纽结和通过10个交叉点的素不可分裂链环的表，除了3个纽结和2个链环之外，所有纽结和链环都可以被证明是环圈交替的。设T是嵌入可定向三维流形M中的环面。设L是M中的一个环，它可以被同构到T的一个邻域T × I中。假设如果T x I收缩到T上，则L投影到T上的连通4-瓣图，使得如果我们跟踪交叉点，当从T的一侧观察时，它们随着链路的分量被遍历而在上和下之间交替。另外，假设T上的每一条非平凡曲线都与L在T上的投影相交。如果一个流形的亏格为一个Heegaard分裂，则有一个唯一的环面直到合痕，它将流形分裂成两个实心环面。(See[3]）因此，我们可以在这些流形中定义一个环面交替的链路，它是关于这个特定环面的环面交替的链路。
THERE have been various generalizations of the class of alternating links in S3, specifically alternative, pseudo-alternating, homogeneous, adequate, augmented alternating and almost alternating links. These generalizations originated out of attempts to extend results known for alternating links to broader classes of links. In this paper, we extend the class of alternating links to a new set, which we call toroidally alternating links. This set of links will be particularly broad, containing within it the set of alternating links, the set of almost alternating links, the subset of augmented alternating links with a single augmenting component, and a sub-class of the set of arborescent links, including all Montesinos links. Surprisingly, if the tables of prime knots through eleven crossings and prime nonsplittable links through ten crossings appearing in [6] are examined, all but three of the knots and two of the links can be shown to be toroidally alternating. Let T be a torus embedded in an orientable 3-manifold M. Let L be a link in M that can be isotoped into a neighborhood T x I of T. Suppose that if T x I is retracted onto T, L projects to a connected 4-valent graph on T such that if one keeps track of the crossings, they alternate between over and under as the components of the link are traversed, when viewed from one side of T. In addition, assume that every nontrivial curve on T intersects the projection of L onto T. Then L is said to be toroidally alternating with respect to T.In the case of a manifold with a genus one Heegaard splitting, there is a unique torus up to isotopy that splits the manifold into two solid tori.(See [3].) Hence, we can define a toroidally alternating link in these manifolds to be a link that is toroidally alternating with respect to this particular torus.