  A knot projection is an image of a generic immersion from a circle into a
two-dimensional sphere. We can find homotopies between any two knot projections
by local replacements of knot projections of three types, called Reidemeister
moves. This paper defines an equivalence relation for knot projections called
weak (1, 2, 3) homotopy, which consists of Reidemeister moves of type 1, weak
type 2, and weak type 3. This paper defines the first non-trivial invariant
under weak (1, 2, 3) homotopy. We use this invariant to show that there exist
an infinite number of weak (1, 2, 3) homotopy equivalence classes of knot
projections. By contrast, all equivalence classes of knot projections
consisting of the other variants of a triple type, i.e., Reidemeister moves of
(1, strong type 2, strong type 3), (1, weak type 2, strong type 3), and (1,
strong type 2, weak type 3), are contractible.
