Finite Type Invariants of Classical and Virtual Knots
Finite Type Invariants of Classical and Virtual Knots
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DOI：
10.1016/s0040-9383(99)00054-3
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发表时间：
1998-10
期刊：
影响因子：
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通讯作者：
M. Goussarov;M. Polyak;O. Viro
中科院分区：
文献类型：
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作者：
M. Goussarov;M. Polyak;O. Viro
我们观察到，任何结不变量扩展到虚拟结。虚拟节点的合痕分类问题被简化为一个代数问题，用箭头图的代数来表示。我们引入了一个新的有限型不变量的概念，并证明了任何这样的不变量的限制度n经典结是一个不变量的度n在经典意义上。通过高斯图公式定义了一个度为n的泛不变量.这种机器是用来获得明确的公式为不变量的低程度。同样的技术也被用来证明任何有限型不变量的经典纽结是由高斯图公式。我们介绍了高斯图的n-等价的概念，并宣布了关于经典n-等价的结果的虚对应部分。
We observe that any knot invariant extends to virtual knots. The isotopy classification problem for virtual knots is reduced to an algebraic problem formulated in terms of an algebra of arrow diagrams. We introduce a new notion of finite type invariant and show that the restriction of any such invariant of degree n to classical knots is an invariant of degree ⩽n in the classical sense. A universal invariant of degree ⩽n is defined via a Gauss diagram formula. This machinery is used to obtain explicit formulas for invariants of low degrees. The same technique is also used to prove that any finite type invariant of classical knots is given by a Gauss diagram formula. We introduce the notion of n-equivalence of Gauss diagrams and announce virtual counter-parts of results concerning classical n-equivalence.