HOW TO TWIST A KNOT

Take a strip of Paper and ''twist'' it, tie a knot on it, and glue its ends together. Then you obtain a closed, twisted, and knotted strip. We use this as as a model for a class of geometric objects which we ca ll the class of closed strips. We define the twisting number of a clos ed strip which is an invariant of ambient isotopy measuring the topolo gical twist of the closed strip. We classify closed strips in euclidea n 3-space by their knots and their twisting number. We prove that this classification exactly divides closed strips into isotopy classes. Us ing this classification we point out how some polynomial invariants fo r links lead to polynomial invariants for strip links. We give a metho d for knotting a strip with control on its twist, and our method inclu des a closed braid description of a closed strip. Finally, we generali ze the notion of closed braids, allowing braids to be closed by any or iented knot and not only by the unknot.