Third-order braid invariants

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Abstract
The author 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 (t) in a two-dimensional configuration space. This configuration space consists of a set of triangular regions connected at their vertices. The curve gamma (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. The author 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.

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