A geometric proof that alternating knots are non-trivial
A geometric proof that alternating knots are non-trivial
复制标题
交替结并非平凡的几何证明
DOI：
10.1017/s0305004100069887
复制
发表时间：
1991
影响因子：
0.8
通讯作者：
M. Thistlethwaite
中科院分区：
文献类型：
--
作者：
W. Menasco;M. Thistlethwaite
关于3-球面上交错的经典链环的非平凡性，文献中有许多证明，但几乎都使用涉及某种代数不变量的组合论证，即行列式[1]、Alexander多项式[3]、Jones多项式[5]，以及在[6]中Brandt-Lickorish-Millett的Q-多项式。事实上，交替链接在这些和其他不变量方面表现得非常好，但这一事实并没有导致对交替链接类型的任何显著的几何理解。因此，寻找这些环的几何性质的纯几何证明是很自然的。Gabai在[4]中给出了一个相关结果的显著几何证明，这一证明也是在[3]中用代数方法证明的，即由Seifert算法从简化的交错链图中得到的Seifert曲面具有该链环的最小亏格。在这里，我们给出了交错纽结非平凡的一个初等几何证明，使用了[7，8]中提出的技巧的一个微小变化。值得注意的是，如果L是一个由多个分量组成的环，而L的某个分量被一个内部位于L补的圆盘所跨越，则L是分裂环，即在S3\L中被一个2球分开，因此我们在这里不考虑多个分量的交替环，正如文献[7]中证明的那样，一个连通的交替图不能表示一个分裂环。
There are many proofs in the literature of the non-triviality of alternating, classical links in the 3-sphere, but almost all use a combinatorial argument involving some algebraic invariant, namely the determinant [1], the Alexander polynomial [3], the Jones polynomial [5], and, in [6], the Q-polynomial of Brandt–Lickorish–Millett. Indeed, alternating links behave remarkably well with respect to these and other invariants, but this fact has not led to any significant geometric understanding of alternating link types. Therefore it is natural to seek purely geometric proofs of geometric properties of these links. Gabai has given in [4] a striking geometric proof of a related result, also proved earlier by algebraic means in [3], namely that the Seifert surface obtained from a reduced alternating link diagram by Seifert's algorithm has minimal genus for that link. Here, we give an elementary geometric proof of non-triviality of alternating knots, using a slight variation of the techniques set forth in [7, 8]. Note that if L is a link of more than one component and some component of L is spanned by a disk whose interior lies in the complement of L, then L is a split link, i.e. it is separated by a 2-sphere in S3\L; thus we do not consider alternating links of more than one component here, as it is proved in [7] that a connected alternating diagram cannot represent a split link.