1 Department of Mathematics, Technical University of Denmark2 Department of Applied Mathematics and Computer Science, Technical University of Denmark
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 a 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 the topological twist of the closed strip. We classify closed strips in euclidean 3-space by their knots and their twisting number. We prove 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 baids, allowing braids to be closed by any oriented knot and not only by the unknot.