Third-order link integrals

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Abstract
The Gauss link integral measures simple linking between two curves. Helicity integrals, which are related to the Hopf invariant, similarly measure the net linking of a set of field lines (for example vortex lines or magnetic lines of force). However, these quadratic integrals do not always detect links involving three or more curves. The author presents an invariant cubic integral which can indeed detect linkage when the quadratic integrals vanish: for example the integral distinguishes the Borromean rings from three unlinked rings. This integral is based on an algebraic topology construct, the Massey trip product.

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