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We consider an analogue of well-known Casson knot invariant for knotoids. Westart with a direct analogue of the classical construction which gives twodifferent integer-valued knotoid invariants and then focus on its homologyextension. Value of the extension is a formal sum of subgroups of the firsthomology group $H_1(\Sigma)$ where $\Sigma$ is an oriented surface with (maybe)non-empty boundary in which knotoid diagrams lie. To make the extensioninformative for spherical knotoids it is sufficient to transform an initialknotoid diagram in $S^2$ into a knotoid diagram in the annulus by removingsmall disks around its endpoints. As an application of the invariants we provetwo theorems: a sharp lower bound of the crossing number of a knotoid (theestimate differs from its prototype for classical knots proved by M.Polyak andO.Viro in 2001) and a sufficient condition for a knotoid in $S^2$ to be aproper knotoid (or pure knotoid with respect to Turaev's terminology). Finallywe give a table containing values of our invariants computed for all sphericalprime proper knotoids having diagrams with at most $5$ crossings.

Homological Casson type invariant of knotoids