We consider an analogue of the well-known Casson knot invariant for knotoids. We start with a direct analogue of the classical construction which gives two different integer-valued knotoid invariants and then focus on its homology extension. The value of the extension is a formal sum of subgroups of the first homology group \(H_1(\varSigma )\) where \(\varSigma \) is an oriented surface with (maybe) non-empty boundary in which knotoid diagrams lie. To make the extension informative for spherical knotoids it is sufficient to transform an initial knotoid diagram in \(S^2\) into a knotoid diagram in the annulus by removing small disks around its endpoints. As an application of the invariants we prove two theorems: a sharp lower bound of the crossing number of a knotoid (the estimate differs from its prototype for classical knots proved by M. Polyak and O. Viro in 2001) and a sufficient condition for a knotoid in \(S^2\) to be a proper knotoid (or pure knotoid with respect to Turaev’s terminology). Finally we give a table containing values of our invariants computed for all spherical prime proper knotoids having diagrams with at most 5 crossings.

我们考虑著名的 Casson 结不变量的类似物。我们从经典构造的直接模拟开始，它给出了两个不同的整数值结节不变量，然后专注于它的同源扩展。扩展的值是第一个同源群\(H_1(\varSigma )\)的子群的形式和，其中\(\varSigma \)是具有（可能）非空边界的定向曲面，其中有节状图。为了使球形结节的扩展信息丰富，在\(S^2\) 中转换初始结节图就足够了通过去除其端点周围的小圆盘，在环中形成一个节状图。作为不变量的应用，我们证明了两个定理：knotoid 的交叉数的尖锐下界（估计不同于 M. Polyak 和 O. Viro 在 2001 年证明的经典结的原型）和一个充分条件knotoid在\（S ^ 2 \）是一个适当的knotoid（或纯knotoid相对于Turaev的术语）。最后，我们给出了一个表格，其中包含我们为所有具有最多 5 个交叉点的图的球形素数固有结点计算的不变量值。