- Werner Burau introduced a twisted geometric mystery in the 1930s by translating braids into matrices, sparking a century-long question about whether these representations were faithful.
- Mathematicians Joan Birman, Tara Brendle, and Vasudha Bharathram resolved the case in 2026, proving that the four-strand braid matrices remain faithful, contrary to assumptions.
- The proof showed that for braids with three or fewer strands, matrices are always faithful; for five or more strands, they are unfaithful; the four-strand case is a critical threshold.
- The team adapted John Moody's geometric method to prove faithfulness, overturning the previous focus on looking for unfaithfulness and solving a major problem in group and knot theory.