= Vertex link in a polygonal schema
{title2=$\operatorname{Lk}(v)$}
Replace each corner of a <polygonal schema> by a small <closed interval> cutting across that corner. The endpoint identifications induced by paired polygon edges glue these <intervals> into a <graph> at each vertex <equivalence class>. This is its vertex link. A <circle> link gives a <disc> <neighbourhood>; a <closed interval> link gives a half-<disc> <neighbourhood> in a <surface with boundary>.
For a complete pairing of the sides of a single polygon, every link endpoint is paired exactly once. In each vertex <equivalence class> the link <graph> is <connected>, since the corner endpoint identifications generate precisely that class, and all its vertices have degree two. It is therefore a <circle>. For partial pairings, a <connected> link can instead be a <closed interval> when it has two unpaired endpoints. Counting free polygon edges alone does not count <boundary components of a surface>; their endpoints must first be identified.
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