On Free Knots and Links
On Free Knots and Links
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发表时间：
2009
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通讯作者：
V. Manturov
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作者：
V. Manturov
经典纽结和虚纽结都是以形式高斯图的形式出现的，其模为对应于Reidmeister运动的一些抽象运动。如果我们忘记了交叉点的结构和扭结的数量模相同的雷德迈斯特移动，我们得到一个戏剧性的简单fi阳离子的虚拟结，它杀死了所有的经典结。然而，在这种简单的fi正离子之后，许多虚节仍然存在。我们构造了这些对象的不变量，并给出了它们在虚节的极小性问题以及与图链相关的问题中的应用。人们可以很容易地将这些结果推广到可定向的情况，并将它们应用于解决不可逆性问题。这些不变量背后的主要思想是一些几何构造，它降低了一般等价于仅模Reidemister-2运动的等价性。本文是文[5]的续篇。1导论在[5]中，对于一类对象(虚结点的Terrific Simplifi正交)，我们证明了一个定理，即某些图的等价问题往往可以归结为一些非常简单的等价问题(仅使用Reidmeister2步)。为了做到这一点，我们在两种类型的交叉点之间建立了差异：“奇数”交叉点和“偶数交叉点”，并创建了我们的对象的图值不变量(自由结)。对于一些“不可约奇数”的图，不变量的值是由图本身组成的，直到第二个Reidomeistermove才被考虑。这有很多推论(已经在这里描述过或仍将在即将发表的论文中出现)：对于虚结点的fl(虚结点及其推广见[4])，它们的非平凡、不等价、不可逆性等。然而，构造的主要神经是奇数交叉的概念。粗略地说，我们拿了一张高斯图，忘记了所有上下的信息和所有以形式为模的雷德迈斯特动作的数字。奇数交叉正好是对应于
AbstractBoth classical and virtual knots arise as formal Gauss diagrams modulo some abstract movescorresponding to Reidemeister moves. If we forget about both over/under crossings structure andwrithe numbers of knots modulo the same Reidemeister moves, we get a dramatic simplificationof virtual knots, which kills all classical knots. However, many virtual knots survive after thissimplification.We construct invariants of these objects and present their applications to minimality problemsof virtual knots as well as some questions related to graph-links.One can easily generalize these results for the orientable case and apply them for solvingnon-invertibility problems.The main idea behind these invariants is some geometrical construction which reduces thegeneral equivalence to the equivalence only modulo Reidemeister - 2 move.This paper is a sequel of the paper [5]. 1 Introduction In [5], for a certain class objects (a terrific simplification of virtual knots) we proved a theoremthat the equivalence questions of some diagrams can often be reduced to the question of some verysimple equivalence (using only Reidemeister 2 moves). To do that, we made difference between twotypes of crossings: the “odd” ones and the “even” ones and created a diagram-valued invariant ofour objects (free knots). For some diagrams which are “irreducibly odd” the value of the invariantconsists of the diagram itself, and it has been considered then only up to second Reidemeistermove.This has a lot of corollaries (already described here or still to come in forthcoming papers):for flat virtual knots (for virtual knots and their generalizations see [4]), their non-triviality,non-equivalence, non-invertibility etc.However, the main nerve of the construction was the notion of odd crossing. Roughly speaking,we are taking a Gauss diagram and forgetting all over/under information and all writhe numbersmodulo formal Reidemeister moves. Odd crossings are precisely those corresponding to