  Quantum phases can be classified by topological invariants, which take on
discrete values capturing global information about the quantum state. Over the
past decades, these invariants have come to play a central role in describing
matter, providing the foundation for understanding superfluids, magnets, the
quantum Hall effect, topological insulators, Weyl semimetals and other
phenomena. Here we report a remarkable linking number (knot theory) invariant
associated with loops of electronic band crossings in a mirror-symmetric
ferromagnet. Using state-of-the-art spectroscopic methods, we directly observe
three intertwined degeneracy loops in the material's bulk Brillouin zone
three-torus, $\mathbb{T}^3$. We find that each loop links each other loop
twice. Through systematic spectroscopic investigation of this linked loop
quantum state, we explicitly draw its link diagram and conclude, in analogy
with knot theory, that it exhibits linking number $(2,2,2)$, providing a direct
determination of the invariant structure from the experimental data. On the
surface of our samples, we further predict and observe Seifert boundary states
protected by the bulk linked loops, suggestive of a remarkable Seifert
bulk-boundary correspondence. Our observation of a quantum loop link motivates
the application of knot theory to the exploration of exotic properties of
quantum matter.
