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.

我们使用两种不同的方法将结的 Vassiliev（或有限类型）不变量理论扩展到结节。首先，我们对knotoids 进行闭包以获得结，我们使用Vassiliev 不变量作为结，证明这些是knotoid 同位素不变量。其次，我们直接在knotoids上定义有限类型不变量，通过Vassiliev skein关系将knotoid不变量扩展到奇异knotoid不变量。然后，对于球形结节，我们表明存在非平凡的 1 类不变量，这与其中 1 类不变量消失的经典结理论形成对比。我们通过对一阶线性弦图进行分类，给出了球形结节的类型 1 不变量的完整理论，并给出了仿射指数多项式和扩展括号多项式的例子。