3RD-ORDER BRAID INVARIANTS

This report analyses the topological invariants of three-braided curves a(t), b(t) and c(t). 3-braids are represented as a single phase curve gamma approximately (t) in a two-dimensional configuration space. This configuration space consists of a set of triangular regions connected at their vertices. The curve gamma approximately (t) passes through a vertex whenever a(t), b(t) and c(t) are collinear. The sequence of vertices completely describes the braid (up to uniform twists). The length T of this sequence can be employed as a measure of topological complexity. The energy of a set of braided magnetic flux tubes is expected to be proportional to T2 + W2, where W is the total winding number (or signed crossing number) of the braid. Second-order winding numbers are integrals of closed 1-forms like d-theta-ab. This report presents a third-order winding number PSI(gamma) which is also an integral of a closed 1-form, but which depends on relations between all three curves. The number PSI(gamma) can be non-zero even when all the second-order winding numbers vanish. Furthermore, PSI(gamma) bears a simple relation to the Massey third-order linking number.