Embedding knots and links in an open book III. On the braid index of satellite links
Embedding knots and links in an open book III. On the braid index of satellite links
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在一本打开的书里嵌入结和链接 III。
DOI：
10.1017/s0305004198002849
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发表时间：
1999
影响因子：
0.8
通讯作者：
Ian J. Nutt
中科院分区：
文献类型：
--
作者：
P. Cromwell;Ian J. Nutt
设V∧S3是一个实心的打结环面。通过Birman和Menasco[2]的工作，观测到卫星链路L=C[midast]P∧V（伴C，模式P和本质环T=∂V）属于两大类别之一：反向弦和非反向弦。这些类别都是[2]中所识别的三种嵌入类型，它们的区别在于是否存在一个有方向的子午盘D∧V，其与有方向的卫星C[midast]P的点相交都有类似的方向。如果不存在这样的D，则称L为反向串卫星。如果C[midast]P是一个非反向串卫星，则已知编织索引b（C[midast]P）仅依赖于b(C)和图案的某些指定属性，而不依赖于分帧参数f∈Z的选择。在[2]中，我们推测，如果C[midast]fP是一个反向串卫星（其中分帧参数为f），则编织指数b（C[midast]fP）取决于弧度指数α(C)、图的性质和分帧。通过建立辫状指数的上界和下界，进一步研究了这种相关性。上界来自L的显式闭合辫状图，给出了其构造。下界来自Homfly多项式，通过Morton-Franks-Williams不等式[5,8]。我们将证明下面的定理。
Let V⊂S3 be a solid, knotted torus. Through the work of Birman and Menasco [2], the observation has been made that a satellite link L=C[midast ]P⊂V (with companion C, pattern P and essential torus T=∂V) falls into one of two broad categories: reverse string and non-reverse string. These categories are borne of the three embedding types identified in [2] and are distinguished by the existence of, or lack of, a meridional disc D⊂V with an orientation, whose point-intersections with the oriented satellite C[midast ]P are all similarly oriented. If no such D exists, then L is said to be a reverse string satellite. If C[midast ]P is a non-reverse string satellite, it is known that the braid index b(C[midast ]P) is dependent only on b(C) and certain specified properties of the pattern and not on the choice f∈Z of framing parameter. In [2] it is conjectured that, if C[midast ]fP is a reverse string satellite (in which the framing parameter is f), the braid index b(C[midast ]fP) depends on the arc index α(C), properties of the pattern and also the framing. We study this dependence further via established upper and lower bounds for braid index. The upper bound comes from explicit closed braid diagrams of L, for which constructions are shown. The lower bound comes from the Homfly polynomial, via the Morton–Franks–Williams inequality [5, 8]. We will prove the following theorem.