Some curves look tangled but can actually be untangled
If you can move the strand around without cutting it and completely straighten it into a smooth circle, it is “not a knot” (in mathematical terms, an unknot). If no amount of moving can ever untangle it, it is a true knot. It can be hard to tell just by looking, so mathematicians have developed reliable tests such as three-coloring.
Rule: At a crossing, the strand with the gap (the small opening you can see in the picture above) is the strand that passes underneath. Decide whether this curve is a knot or not. If there are fewer than 3 crossings, it can always be untangled, no matter how the curve is drawn — this is a mathematically proven fact!
Question 1/4 · Correct 0
This trefoil knot is made of 3 strands. Click a strand to change its color.
Three-coloring rule: Use all 3 colors and color the strands so that at every crossing, the three strands meeting there are either “all the same color” or “all different colors.” (Using only one color does not count!) Interestingly, the trefoil succeeds, while an unknot (a circle) has only one strand, so there is no way for it to satisfy this rule in an interesting way in the first place — that is why being “three-colorable” can serve as evidence that something is a true knot.
💡 Three-coloring is one of the simplest examples of a “knot invariant,” something mathematicians actually use to distinguish knots. If two knot diagrams differ in whether they can be three-colored, then no matter how similar they look, they can never represent the same knot.