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Quantum phases can be classified by topological invariants, which take ondiscrete values capturing global information about the quantum state. Over thepast decades, these invariants have come to play a central role in describingmatter, providing the foundation for understanding superfluids, magnets, thequantum Hall effect, topological insulators, Weyl semimetals and otherphenomena. Here we report a remarkable linking number (knot theory) invariantassociated with loops of electronic band crossings in a mirror-symmetricferromagnet. Using state-of-the-art spectroscopic methods, we directly observethree intertwined degeneracy loops in the material's bulk Brillouin zonethree-torus, $\mathbb{T}^3$. We find that each loop links each other looptwice. Through systematic spectroscopic investigation of this linked loopquantum state, we explicitly draw its link diagram and conclude, in analogywith knot theory, that it exhibits linking number $(2,2,2)$, providing a directdetermination of the invariant structure from the experimental data. On thesurface of our samples, we further predict and observe Seifert boundary statesprotected by the bulk linked loops, suggestive of a remarkable Seifertbulk-boundary correspondence. Our observation of a quantum loop link motivatesthe application of knot theory to the exploration of exotic properties ofquantum matter.

What's knot to like? Observation of a linked loop quantum state