Fun with Knots

Some mathematicians (like Dr. Lisa Piccirillo) study objects called knots. Knots are any object that can be made by taking a piece of string, tangling it around itself, and sticking the ends together. In this activity, you’ll tie some knots and explore how to tell when two knots that look different are actually the same. You’ll need 6 pieces of string. A good length for each piece is approximately 1 foot long, but this doesn’t need to be exact.


A knot projection is a drawing of a knot on 2D paper. Starting with your string untied, see if you can construct the knots drawn below out of string by first tangling the string up and then tying the ends together.



Picture of six knots drawn in green, with three in the top row and three in the bottom row.
Some knots you can construct.

Drawing of the unknot (which looks like a circle) in bright green.
The unknot is a loop with no tangles.

It turns out the knots in the bottom row of the image above can all be turned into the unknot (a loop with no tangles) by continuous deformations (stretching, shrinking, twisting, and bending). See if you can turn these three knots into the unknot using only continuous deformations, without untying the ends.


There are also more complicated knots that can be turned into the unknot with the same types of deformations! See this video for an example of untangling a very complicated unknot.


Image of a green trefoil knot.

Now try to untangle the knots in the top row of the first image. No matter how long or hard you try, you won’t be able to (read more about why here). You can, however, turn the knot on the top right into another knot called the trefoil knot, shown on the left. See if you can figure out how!