since the open end means that they can be shrunk to a point), so a possibly disconnected compact 1-manifold is a collection of n intervals and m circles The condition that the boundary of X lies in says that intervals either connect two lines or connect two points on one of the lines, but imposes no conditions on the circles. One may view tangles as having a vertical direction (I), lying between and possibly connecting two lines ( and ), and then being able to move in a two-dimensional horizontal direction () between these lines; one can project these to form a tangle diagram, analogous to a knot diagram. Tangles include links (if X consists of circles only), braids, and others besides – for example, a strand connecting the two lines together with a circle linked around it. In this context, a braid is defined as a tangle which is always going down – whose derivative always has a non-zero component in the vertical (I) direction. In particular, it must consist solely of intervals, and not double back on itself; however, no specification is made on where on the line the ends lie. A string link is a tangle consisting of only intervals, with the ends of each strand required to lie at (0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (2, 1), ... – i.e., connecting the integers, and ending in the same order that they began (one may use any other fixed set of points); if this has ℓ components, we call it an "ℓ-component string link". A string link need not be a braid – it may double back on itself, such as a two-component string link that features an overhand knot. A braid that is also a string link is called a pure braid, and corresponds with the usual such notion. The key technical value of tangles and string links is that they have algebraic structure. Isotopy classes of tangles form a tensor category, where for the category structure, one can compose two tangles if the bottom end of one equals the top end of the other (so the boundaries can be stitched together), by stacking them – they do not literally form a category (pointwise) because there is no identity, since even a trivial tangle takes up vertical space, but up to isotopy they do. The tensor structure is given by juxtaposition of