Unbelievable Unlinking

Imagine that the two objects in Figure 1 are solid (with thickness) and made of very flexible and stretchy rubber.

Question: is it possible to deform one object into the other in a continuous motion (without tearing or cutting)?

Surprise answer: Yes!!

Hint: it is important that the object is solid and has thickness; this transformation cannot be done with a one-dimensional piece of string. It is also not possible to do this with a piece of rope because even though the rope has thickness, it is not flexible or “stretchy” enough. See below for an explanation and animated gif. Or, don’t scroll down if you want to think about it a while!

Presentation Suggestions:
Students (as well as you) may find this very hard to believe! If you like this one, see also Pretzel Unlinking.

The Math Behind the Fact:
One way to do this is the following. Widen one of the loops and move one of its handles along the stem between the two loops to the other loop and push it through the hole so that the two loops become unlinked. The reference contains a sequence of pictures of this transformation.

Graeme McRae has generously contributed the animated gif in Figure 2, showing another solution to this problem! (Thank you, Graeme!)

You can take a course in topology to learn more about properties of objects that do not change under continuous deformations.

How to Cite this Page: 
Su, Francis E., et al. “Unbelievable Unlinking.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

V.V. Prasalov, Intuitive Topology.

Fun Fact suggested by:
Francis Su, Graeme McRae

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