Link genus and the Conway moves
Link genus and the Conway moves
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链接属和康威移动
DOI：
10.1007/bf02564693
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发表时间：
1989
影响因子：
0.9
通讯作者：
A. Thompson
中科院分区：
文献类型：
--
作者：
M. Scharlemann;A. Thompson
因此，特别地，V+， V_和zV0中至少有两个具有相同的度，且不小于第三个的度。3流形中定向连杆L的Seifert曲面是一个紧致定向曲面，其边界为连杆，其所有的分量都不闭合。定义x (L)为L的所有Seifert曲面的最大欧拉特征。如果L是S a中的非分裂交替连杆，则deg (VL)= 1—x (L)[Cr]。因此，如果L+， L_和Lo都是非分裂的交替链接，则x (L+), x （L-）和x (Lo)-1中的两个相等且不大于第三个。我们将证明这个关系对于任意链接都是成立的。两个结果是：a)链路L的Conway绞结图的高度以x (L)为界。特别是，这为计算新的定向结多项式的复杂度提供了一个意想不到的下界。B)双重结是指结的属数和解结数都是1的结。
Hence in particular, at least two of V+, V_, and zV0 have the same degree, which is no smaller than the degree of the third. A Seifert surface for an oriented link L in a 3-manifold is a compact oriented surface none of whose components are closed and whose boundary is the link. Define x (L) to be the maximal Euler characteristic of all Seifert surfaces for L. If L is a non-split alternating link in S a then deg (VL)= 1--x (L)[Cr]. Hence if L+, L_ and Lo are all non-split alternating links, then two of x (L+), x (L-) and X (Lo)-1 are equal and are no larger than the third. We will show that this relation remains true for arbitrary links. Two consequences are: a) the height of the Conway skein diagram for a link L is bounded below by-x (L). In particular, this gives an unexpected lower bound for the complexity of calculating the new oriented knot polynomials. b) doubled knots are precisely those knots whose genus and unknotting number are both 1.