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On Some Moves on Links and the Hopf Crossing Number
Mediterranean Journal of Mathematics ( IF 1.1 ) Pub Date : 2020-11-16 , DOI: 10.1007/s00009-020-01611-6
Maciej Mroczkowski

We consider arrow diagrams of links in \(S^3\) and define k-moves on such diagrams, for any \(k\in \mathbb {N}\). We study the equivalence classes of links in \(S^3\) up to k-moves. For \(k=2\), we show that any two knots are equivalent, whereas it is not true for links. We show that the Jones polynomial at a k-th primitive root of unity is unchanged by a k-move, when k is odd. It is multiplied by \(-1\), when k is even. It follows that, for any \(k\ge 5\), there are infinitely many classes of knots modulo k-moves. We use these results to study the Hopf crossing number. In particular, we show that it is unbounded for some families of knots. We also interpret k-moves as some identifications between links in different lens spaces \(L_{p,1}\).



中文翻译:

关于链接的某些移动和霍夫夫穿越数

我们考虑\(S ^ 3 \)中链接的箭头图,并为任何\(k \ in \ mathbb {N} \)中的图定义k移动。我们研究\(S ^ 3 \)中直到k个运动的链接的等价类。对于\(k = 2 \),我们表明任意两个结都是等效的,而对于链接则不是这样。我们表明,当k为奇数时,在第k个本原根处的Jones多项式不会因k-移动而改变。当k为偶数时,它乘以\(-1 \)。由此可见,对于任何\(k \ ge 5 \),都存在无限多类以k为模的结-移动。我们使用这些结果来研究霍普夫穿越数。特别是,我们表明它对于某些系列的结是不受限制的。我们还将k移动解释为不同镜头空间\(L_ {p,1} \)中的链接之间的某些标识。

更新日期:2020-11-16
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