Classification of Closed Strips in the Three Dimensional Euclidean Space

Take a strip of paper and
'twist' it, tie a knot on it,
and glue its ends together.
This is the model for a class
of geometric objects which we
call the class of closed
strips. We define the
twisting number of a closed
strip which is an invariant
of ambient isotopy measuring
its topological twist. We
classify closed strips in
euclidean 3-space by their
knots and their twisting
number. We have proven that
this classification exactly
divides closed strips into
isotopy classes. Using this
classification we point out
how some polynomial
invariants for links lead to
polynomial invariants for
strip links. We give a method
for knotting a strip with
control on its twist, and our
method includes a closed
braid description of a closed
strip. Finally, we generalize
the notion of closed braids,
allowing braids to be closed
by any oriented knot and not
only by the unknot. The
inverse braid closing
operator problem is still
open, but it contains Markovs
Theorem for classical closed
braids as a special case.