Complementary Regions of Knot and Link Diagrams
Complementary Regions of Knot and Link Diagrams
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DOI：
10.1007/s00026-011-0109-2
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发表时间：
2008-12
影响因子：
0.5
通讯作者：
C. Adams;Reiko Shinjo;Kokoro Tanaka
中科院分区：
文献类型：
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作者：
C. Adams;Reiko Shinjo;Kokoro Tanaka
对于结点和链环，如果每个结点和链环在球面上都有一个简化的投影，使得投影的每个补面的边数来自给定的序列，则称递增的整数序列对于节点和链环是通用的。本文证明了下列无穷序列对结点和链环都是通用的：(3，5，7，.。.)，(2，n，n+n 1，n+n 2，.。。)对于每个n≥3，(3，n，n+n 1，n+n 2，.。。)对于每个≥4，每个≥5的有限序列(2，4，5)和(3，4，n)对所有的结和环都是通用的。还证明了每个纽结都有一个恰好有两个奇边面的投影，这可以看作是三角形，n个分支的每个环都有一个投影，如果是偶数则有最多的节点边面，如果是奇数则有n+1个奇边面。
An increasing sequence of integers is said to beuniversalfor knots and links if every knot and link has a reduced projection on the sphere such that the number of edges of each complementary face of the projection comes from the given sequence. In this paper, it is proved that the following infinite sequences are each universal for knots and links: (3, 5, 7, . . .), (2,n,n+ 1,n+ 2, . . .) for eachn≥ 3, (3,n,n+ 1,n+ 2, . . .) for eachn≥ 4. Moreover, the finite sequences (2, 4, 5) and (3, 4,n) for eachn≥ 5 are universal for all knots and links. It is also shown that every knot has a projection with exactly two odd-sided faces, which can be taken to be triangles, and every link ofncomponents has a projection with at mostnodd-sided faces ifnis even andn+ 1 odd-sided faces ifnis odd.