Daily Archives: May 29, 2026

2026-08 Crossing-Change Paths from the Trefoil to the Figure-Eight Knot

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.)

Solution: 2026-06 Polynomial integrals

Let \(f(x)\) be a function such that \((1-x^2) f”(x) – 2x f'(x) + \alpha (\alpha+1) f(x) =0\)
for some \(\alpha \not\in \mathbb{N}\). Define \(P_n (x) = \frac{d^n}{dx^n} (x^2-1)^n\) for \(n =0,1,…\). Compute \(\int_{-1}^1 f(x) P_n(x) dx.\)

The best solution was submitted by 정서윤 (수리과학과 23학번, +4). Congratulations!

Here is the best solution of problem 2026-06.

Other solutions were submitted by 김은성 (서울대 수리과학부, +3), 신민규 (수리과학과 24학번, +3), 이상주 (경남대 수학교육과, +3), 장현준 (서울과학고 3학년, +3), 지은성 (수리과학과 석박통합과정, +3), Huseyn Ismayilov (전산학부 22학번, +3).