Non-triviality of the Jones polynomial and the crossing numbers of amphicheiral knots
Non-triviality of the Jones polynomial and the crossing numbers of amphicheiral knots
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琼斯多项式的非平凡性和两栖结的交叉数
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
A. Stoimenow
中科院分区：
文献类型：
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作者：
A. Stoimenow
通过对Jones多项式的深入研究，我们确定了(素数)双面纽结的交叉数，作为我们的主要结果。作为进一步的应用，我们证明了几类环，包括半闭环和半闭环的怀特黑德偶，都具有非平凡的Jones多项式。我们还证明了存在无穷多个没有正极小交叉图的正纽结。给出了环的扭数、Mahler测度和双曲体积的一些关系，例如Montesinos环和3-辫环的体积的上界是用Jones多项式表示的。
Using an involved study of the Jones polynomial, we determine, as our main result, the crossing numbers of (prime) amphicheiral knots. As further applications, we show that several classes of links, including semiadequate links and Whitehead doubles of semiadequate knots, have non-trivial Jones polynomial. We also prove that there are infinitely many positive knots with no positive minimal crossing diagrams. Some relations to the twist number of a link, Mahler measure and the hyperbolic volume are given, for example explicit upper bounds on the volume for Montesinos and 3-braid links in terms of their Jones polynomial.