Classification of Closed Strips in the Three Dimensional Euclidean Space

    Project Details

    Description

    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.
    StatusFinished
    Effective start/end date01/01/199601/06/1997