Trefoil to Figure-Eight by Crossing Changes

A knot is a simple closed curve in three-dimensional space. A diagram of a knot is a projection of the knot to the plane together with over/under information at each crossing. A crossing change is the operation of switching one crossing from over to under or from under to over.

Find up to three examples of sequences of knot diagrams that begin with a diagram of the trefoil knot and end with a diagram of the figure-eight knot, where each step is obtained from the previous diagram by a crossing change.

The obvious chains trefoil → unknot → figure-eight and trefoil → trefoil # figure-eight → figure-eight are not allowed.

(3 points will be given for three correct examples, 2 points for two correct examples, and 1 point for one correct example.)

The best solution was submitted by 이원준 (Rutgers University, +4). Congratulations!

Here is the best solution of problem 2026-08.

Other solutions were submitted by 신민규 (수리과학과 24학번, +3), 정서윤 (수리과학과 23학번, +3), 장현준 (서울과학고 3학년, +2).

Let
\[
Q_n=\{0,1\}^n
\]
be the \(n\)-dimensional discrete cube, viewed as a graph in which two vertices are adjacent if they differ in exactly one coordinate.

For a subset \(A\subseteq Q_n\), let \(\partial_e A\) denote the set of edges with one endpoint in \(A\) and the other in \(Q_n\setminus A\).

Prove that for every \(A\subseteq Q_n\),
\[
|\partial_e A|\ge |A|\bigl(n-\log_2|A|\bigr).
\]

The best solution was submitted by 김지원 (전산학부 24학번, +4). Congratulations!

Here is the best solution of problem 2026-07.

Other solutions were submitted by 김은성 (서울대 수리과학부, +3), 신민규 (수리과학과 24학번, +3), 장현준 (서울과학고 3학년, +3), 정서윤 (수리과학과 23학번, +3), 최완수 (서울대 물리천문학부, +3).