论文标题
在$ n $ n $ dripialities of Classical和virtual结中
On $n$-trivialities of classical and virtual knots for some unknotting operations
论文作者
论文摘要
在本文中，我们引入了一种新的非平凡过滤，称为F-order，用于古典和虚拟结。这种过滤会产生过滤的结不变性，称为有限类型不变式，类似于Vassiliev结的不变性。由Goussarov，Polyak和Viro引入的有限类型不变性是众所周知的，我们称它们为GPV订单的有限类型不变性。我们表明，对于任何积极的整数$ n $，对于任何经典的打结$ k $，都存在无限的许多非平凡的经典结，所有的gpv订单$ <n $ varemiants均与$ k $（Theorem 1）相吻合。此外，我们表明，对于任何积极的整数n，都存在一个非平凡的虚拟结，其F级$ <n $的有限型不变性与琐碎结（定理2）相吻合。为了证明定理1（定理2，分别），我们通过某个无结的操作（称为虚拟化（禁止移动，分别））定义了n-平地，并且对于任何正整数n，找到一个N-琐事的经典结（虚拟结，virtual knot，spess。）。
In this paper, we introduce a new nontrivial filtration, called F-order, for classical and virtual knot invariants; this filtration produces filtered knot invariants, which are called finite type invariants similar to Vassiliev knot invariants. Finite type invariants introduced by Goussarov, Polyak, and Viro are well-known, and we call them finite type invariants of GPV-order. We show that for any positive integer $n$ and for any classical knot $K$, there exist infinitely many of nontrivial classical knots, all of whose finite type invariants of GPV-order $< n$, coincide with those of $K$ (Theorem 1). Further, we show that for any positive integer n, there exists a nontrivial virtual knot whose finite type invariants of our F-order $< n$ coincide with those of the trivial knot (Theorem 2). In order to prove Theorem 1 (Theorem 2, resp.), we define an n-triviality via a certain unknotting operation, called virtualization (forbidden moves, resp.), and for any positive integer n, find an n-trivial classical knot (virtual knot, resp.).
