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.)
The best solution was submitted by 신민규 (수리과학과 24학번, +4). Congratulations!
Here is the best solution of problem 2026-05.
Other solutions were submitted by 장현준 (서울과학고 3학년, +3), 정서윤 (수리과학과 23학번, +2).