Equality of ribbon numbers for symmetric ribbon knots of ribbon number two

Let KK be a symmetric ribbon knot. Its ribbon number r(K)r(K) is the minimum number of ribbon singularities in any ribbon disk bounded by KK, and its symmetric ribbon number rs(K)r_s(K) is the minimum number of ribbon singularities in any symmetric ribbon disk bounded by KK.

Ribbon-number-two conjecture. If r(K)=2r(K)=2, then

rs(K)=2.r_s(K)=2.

All examples in the authors' data with rs(K)>r(K)r_s(K)>r(K) have r(K)3r(K)\geq 3, so the conjecture isolates the first case in which the two invariants might necessarily agree.

Sources & referencesView supporting material

Primary source

Sajid Raihan Akash, Eric Corrado, Bishop Placke, Sam Sanketh, Nick Starns, Anok Timothy and Alexander Zupan, “Symmetric ribbon numbers of low-complexity knots”, arXiv:2606.02390 (2026).

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