Pointed Quandles of Linkoids

In this paper we define the fundamental quandle of knotoids and linkoids and prove that it is invariant under the under forbidden-move and hence encodes only the information of the underclosure of the knotoid. We then introduce n-pointed quandles, which generalize quandles by specifying n elements as ordered basepoints. This leads to the notion of fundamental pointed quandles of linkoids, which enhances the fundamental quandle. Using 2-pointed quandle allows us to distinguish 1-linkoids with equivalent underclosures and leads to a couple of 1-linkoid invariants. We also generalize the notion of homogeneity of quandles to n-homogeneity of quandles. We classify all ∞-homogeneous, finite quandles.