THE 3-MOVE CONJECTURE FOR 5-BRAIDS
THE 3-MOVE CONJECTURE FOR 5-BRAIDS
复制标题
5 辫子的 3 步猜想
DOI：
10.1142/9789812792679_0004
复制
发表时间：
2000
期刊：
影响因子：
--
通讯作者：
Qi Chen
中科院分区：
文献类型：
--
作者：
Qi Chen
1981年，Nakanishi猜想任何一个环都是3-等价于平凡环，J.Przytycki证明了包括11个交叉环在内的某些类环是3-等价的平凡环。在此基础上，我们证明了所有的闭4-辫子都是3-等价于平凡环的。然后，我们证明了所有不超过12个交叉点的链路都是3-等价于平凡链路。对于封闭的5-辫子，只有7个(达到共轭等级)有12个以上的交叉点。我们手工减少了其中的五个。剩下的两个是3-等价的。因此，只有一个未解决的情况，那就是有20个交叉点的5-辫子，如图15所示。我们也证明了类似的结果对于t3，-移动。
Nakanishi conjectured in 1981 that any link is 3-equivalent to a trivial link.J. Przytycki proved that some classes of links, including 11 crossing links, are 3-equivalent to trivial links . Based on this result we prove that all closed 4-braids are 3-equivalent to trivial links. Then we show that all links with no more than 12 crossings are 3-equivalent to trivial links. For closed 5-braids only 7 of them (up to conjugacy class) have more than 12 crossings. We reduce five of them by hand. The remaining two are 3-equivalent. Hence there is only one unsettled case which is a 5-braid with 20 crossings as shown in Figure 15. We prove also similar results for t3,-moves.