Strong Bernhard–Jablan status of the knot 13n1587

Determine whether the knot 13n1587 satisfies the strong Bernhard–Jablan equality $u(13n1587)=u^s_{\mathrm{BJ}}(13n1587)$, equivalently whether some crossing change in a minimal diagram of 13n1587 yields a knot of unknotting number one.

Background

The paper determines that 13n1587 has unknotting number two, while its weak Bernhard–Jablan number is three. This establishes failure of the weak equality but does not decide the strong equality, which depends on whether a minimal-diagram crossing change produces a knot with unknotting number one.

References

This determines the unknotting number of Brittenham and Hermiller's second example, for which the weak equality was already known to fail. Its status under the strong equality depends on whether a crossing change in some minimal diagram gives a knot of unknotting number one; this remains undetermined.

— Computation of unknotting numbers: which knot breaks the Bernhard-Jablan Conjecture  (2609.09861 - Lee, 9 Sep 2026) in Section 3.1, An Explicit Counterexample; Section 4, Conclusion