I’ve compiled a list of some cool activities which are related to topology and geometry, which I envision could be used for public outreach or other recreational math events. Many of the activities can be adjusted to various levels of difficulty, and I’ve tried to give a brief description of each one with links to more detailed resources. If you know of anything that should be added to this list, I would love to hear about it!

Hands-on activities

• Conway’s Rational Tangles: This is a great low-floor high-ceiling exercise which is accessible to anyone who knows about fractions, but actually involves some pretty sophisticated mathematics. You and three friends “tangle” two ropes by performing specific movements, and then try to get back to the initial “untangled” position using the same movements. It turns out that each tangle can be associated with a number and each movement associated with an arithmetic operation, so no matter how complicated the tangles might get, you can always get back to 0 (the untangled position) using the right sequence of movements. There are lots of resources about this activity on the internet, and one of my favorites can be found here.
• Picture frame puzzles: Can you hang a picture frame on two nails such that whenever one of the nails is removed, the picture comes crashing to the ground? Get some cardboard, thumb tacks, and string to test this out! This puzzle, originally posed by A. Spivak in the 1990’s, has some interesting connections to knot theory and group theory. You can also create different versions of the puzzle, e.g. can you hang a picture frame on 3 nails such that whenever any two of them are removed the picture falls, but the picture remains hanging if only one is removed? This paper covers some cool generalizations of the picture frame puzzle, and also goes into a lot of the mathematics behind the various solutions.
• Rope handcuffs (suggested by Andres Mejia): You and a friend tie ropes around your wrists, forming (loose!) handcuffs, which are linked together in a certain way. How can you separate yourself form your partner without breaking the rope or taking the handcuffs off? Here’s a video which shows you how to do it.
• Topological clothing: Can you turn your pants inside out without taking your feet off the ground? Can you turn your shirt inside out with your wrists tied together (and without separating your wrists)? Note that you’ll need pretty roomy clothes to make these work. See how the pants puzzle works in this video (around the 8 minute mark). I couldn’t find a video of the shirt puzzle but it’s pretty similar to the pants puzzle.  Also, this math stackexchange post has some great discussion about the topology of turning clothing inside out. Like the rope handcuffs, this activity is a great way to get comfortable with thinking topologically!
• Mathematically correct breakfast: Have you ever wished you could cover your bagel with slightly more cream cheese? Then you should follow this method of cutting your bagel into two linked halves. You can even use calculus to figure out how much more cream cheese you get (assuming you spread it evenly, of course). This is a hands-on constructive exercise which teaches people about cool topological shapes (torus knots). Plus you get to eat the bagel at the end.
• Two potatoes (suggested by Julian Gould): Given two potatoes, can you draw a closed loop on each so that both loops are the same shape in 3D space? You could go get some potatoes and try this out, or just do this as a brain puzzle. Be warned: this is a puzzle where physical intuition gets in the way of an elegant solution. I won’t spoil it for you, but you can find the solution in the replies of this tweet if you scroll long enough. A more mathematical formulation of the puzzle is discussed in this math overflow post.
• Amazing Rope Trick (suggested by Kyle Ormsby): This one is a bit more of a demonstration rather than an interactive activity, and it’s similar to the rope handcuffs in spirit. You can follow this demonstration to mystify your friends by magically pulling a rope free of two interlocked carabiners. The math behind this trick is explained in more detail in the post I linked, but the basic idea is that the fundamental group of the wedge of two circles is the non-Abelian free group on two generators, while the fundamental group of the torus is the free Abelian group on two generators!

Other resources

• The National Museum of Mathematics ran a weekly column with Make: Online called Math Monday which covers a variety of artsy-math things, some of which could be made into activities.
• Check out Moira Chas’s list of math outreach ideas. Some of the activities on my list were taken from this page!
• Science World has a list of short activities related to topology, quite a few of which involve constructing cool shapes using paper.