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The paradigmatic example of a topological phase of matter, the two-dimensional Chern insulator^{, , , –}, is characterized by a topological invariant consisting of a single integer, the scalar Chern number. Extending the Chern insulator phase from two to three dimensions requires generalization of the Chern number to a three-vector^,, similar to the three-dimensional (3D) quantum Hall effect^{, , , , –}. Such Chern vectors for 3D Chern insulators have never been explored experimentally. Here we use magnetically tunable 3D photonic crystals to achieve the experimental demonstration of Chern vectors and their topological surface states. We demonstrate Chern vector magnitudes of up to six, higher than all scalar Chern numbers previously realized in topological materials. The isofrequency contours formed by the topological surface states in the surface Brillouin zone form torus knots or links, whose characteristic integers …