For every knot K, there exists a minimal-crossing diagram D of K and a crossing of D such that changing that crossing produces a knot K' satisfying u(K')=u(K)-1, where u denotes the unknotting number.

Bernhard–Jablan conjecture

The Bernhard–Jablan conjecture asserts that every knot has a minimal-crossing diagram whose one-crossing change reduces its unknotting number. Brittenham and Hermiller reported a counterexample, making this a negative-resolution problem.

Known results

Brittenham and Hermiller (2017): $K13n3370$ has $u(K13n3370)\le 2$ , while its weak Bernhard–Jablan unknotting number is $3$ .
Their exhaustive computations identify $K12n288$ , $K12n491$ , $K12n501$ , and $K13n3370$ as obstructions; they were independently checked with Knotscape.

September 2026 computational report

On September 9, 2026, Computation of unknotting numbers: which knot breaks the Bernhard–Jablan Conjecture reported $2,525$ previously unknown unknotting numbers and exhibited $13n3370$ as a counterexample. This reinforces the negative claim but does not, in this scan, supply independent verification.

Current status (as of September 2026): The conjecture is reported false via $13n3370$ , with no later repair found, but the computational resolution remains unverified in this report.

Bernhard–Jablan conjecture

Primary source

Additional references

Progress summary

Solutions 0