"KNOT DIVIDED"

Can a  DIVIDED KNOT  be  NOT DIVIDED ?

We start with the simplest possible knot: the overhand knot, also known as the trefoil or pretzel knot,
-- which we then split lenghtwise along the whole strand that forms the three loops.

But there is a twist that may lead to surprises: The knotted strand is actually a triply twisted Moebius band!

Thus the question:
Does our cut separate the structure into two pieces, or does it form a single, highly knotted twisted strand?

FOR MATHEMATICIANS ONLY:

There is a self-referential beauty in our sculpture:
If one forms a Moebius band by twisting a belt through three half-turns (instead of just one),
then the band's edge forms a trefoil knot.

Mathematicians classify the complexity of knots by the minimum number of line crossings
that one must accept when trying to draw that knot on a piece of paper.
The trefoil knot is the simplest knot and cannot be drawn with less than three crossings.
Our sculpture starts out as a triply twisted Moebius band, knotted into a trefoil knot.
When we split it lengthwise, it is still a single knot, but of higher complexity.
Can you figure out what its crossing number is?