  In 2010, Turaev introduced knotoids as a variation on knots that replaces the
embedding of a circle with the embedding of a closed interval with two
endpoints which here we call poles. We define generalized knotoids to allow
arbitrarily many poles, intervals, and circles, each pole corresponding to any
number of interval endpoints, including zero. This theory subsumes a variety of
other related topological objects and introduces some particularly interesting
new cases. We explore various analogs of knotoid invariants, including height,
index polynomials, bracket polynomials and hyperbolicity. We further generalize
to knotoidal graphs, which are a natural extension of spatial graphs that allow
both poles and vertices.
