Kauffman's polynomial and alternating links
Kauffman's polynomial and alternating links
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考夫曼多项式和交替链接
DOI：
10.1016/0040-9383(88)90012-2
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发表时间：
1988
期刊：
影响因子：
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通讯作者：
M. Thistlethwaite
中科院分区：
文献类型：
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作者：
M. Thistlethwaite
这些定理导致快速测试，这将经常区分具有相同交叉数的交替图的链路;具有不同交叉数的简化交替图的链路由[12]的推论1自动区分（也在[4]和[9]中证明）。而且，正如LH Kauffman所指出的，从定理1可以得出，任何一个双螺旋交错纽结的约化交错图都必须有扭曲0。K. Murasugi在[lo]中独立地得到了定理1的一个不同的证明;他发现了一个公式，该公式将琼斯多项式VL（t）中t的极幂与L的约化交错图的扭曲和L的签名联系起来。我感谢WBR Lickorish做出了以下重要的观察：由于F1（u，z）决定VL（t）（见[7]），定理1的这两个证明的组合产生了这样的结果，对于交替链路L，F1（a，z）决定签名。多项式Fi（a，z）通常不确定签名，这由结9和它的反面证明：这些结共享相同的考夫曼多项式，但具有不同的签名。
These theorems result in quick tests which will often distinguish between links having alternating diagrams with the same number of crossings; links having reduced alternating diagrams with different numbers of crossings are automatically distinguished by Corollary 1 of [12](proved also in [4] and [9]). Also, as LH Kauffman has pointed out, it follows from Theorem 1 that any reduced alternating diagram of an amphicheiral alternating knot must have writhe 0. K. Murasugi, in [lo], has independently obtained a different proof of Theorem 1; he has discovered a formula relating the extreme powers of t in the Jones polynomial VL (t) with the writhe of the reduced alternating diagram of L and the signature of L. I am grateful to WBR Lickorish for making the following important observation: since F,(u, z) determines VL (t)(see [7]), the combination of these two proofs of Theorem 1 yields the result that, for alternating links L, F,(a, z) determines the signature. That the polynomial F,(a, z) does not determine the signature in general is evidenced by the knot 9,, and its obverse: these knots share the same Kauffman polynomial, yet have different signatures.