  Knotoids were introduced by V. Turaev as open-ended knot-type diagrams that
generalize knots. Turaev defined a two-variable polynomial invariant of
knotoids which encompasses a generalization of the Jones knot polynomial to
knotoids. We define a triply-graded homological invariant of knotoids
categorifying the Turaev polynomial, called winding homology. Forgetting one of
the three gradings gives a generalization of the Khovanov knot homology to
knotoids.
