论文标题
有限类型的结节
Finite type invariants for knotoids
论文作者
论文摘要
我们使用两种不同的方法扩展了打结的Vassiliev（或有限类型）不变的理论。首先，我们在结中闭合以获得结，然后使用Vassiliev不变性来打结，证明它们是结节的同位素不变的。其次，我们直接通过Vassiliev skein Relation将结节不变性扩展到奇异的结节不变，将有限型不变式定义。然后，对于球形结，我们表明存在非平凡的类型1不变性，与经典结理论相比，其中1型不变性消失。我们通过对One阶的线性和弦图进行分类，对球形结的类型1不变式提供了完整的理论，我们提出了由仿射指数多项式和扩展的支架多项式引起的示例。
We extend the theory of Vassiliev (or finite type) invariants for knots to knotoids using two different approaches. Firstly, we take closures on knotoids to obtain knots and we use the Vassiliev invariants for knots, proving that these are knotoid isotopy invariant. Secondly, we define finite type invariants directly on knotoids, by extending knotoid invariants to singular knotoid invariants via the Vassiliev skein relation. Then, for spherical knotoids we show that there are non-trivial type-1 invariants, in contrast with classical knot theory where type-1 invariants vanish. We give a complete theory of type-1 invariants for spherical knotoids, by classifying linear chord diagrams of order one, and we present examples arising from the affine index polynomial and the extended bracket polynomial.
