Most scientific discoveries justify their significance by filling holes between existing ideas and arguments. Representing knowledge as high-dimensional concept embeddings, we use persistent homology to detect scientific holes of increasing order, from lower-order gaps between disconnected ideas to higher-order cavities within unevenly connected ones; then we identify the research that fills them. Researchers who close holes draw outsized collective attention, staging novelty that the citing literature rewards, most strongly in discovery-oriented fields and least in design-oriented fields. Closing a hole is followed by persistent work on the same combinations of concepts at many times the rate of comparable holes left open. Yet as knowledge grows, higher-order holes explode while the fraction of research that fills them collapses, leaving most higher-order combinations unexplored. These results call for a richer science of holes and mark a frontier where contemporary AI might help fill the gaps human science opens.

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