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Problem of the week

2026-05 Separating a 2-Component Link by Surfaces

A link in S3 is a smooth embedding of a finite disjoint union of circles into S3. A link diagram is a generic projection to S2 together with over/under data at each double point. For an oriented 2-component link K ∪ J, the linking number lk(K, J) is one-half of the signed sum of the crossings between K and J.

Prove or disprove that if lk(K, J) = 0, then there exist disjoint, compact, properly embedded, orientable surfaces F1, F2 ⊂ S3 × I such that

∂F1 = K × {1}
∂F2 = J × {1}.

Your solution should consist almost entirely of pictures. Each picture may have at most one short explanatory sentence.

(It turns out that the converse is also true.)

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