The Invertible Trefoil
When is a knot invertible ?
Age
14 to 16
Challenge level
Problem
If you make a knot from plastic pipe you could imagine filling the pipe with a fluid such as water. The fluid would then be able to flow around the pipe in some direction. In the video we've tied a pipe into a trefoil knot and stencilled some arrows on its surface so you can see the direction of flow.
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Notice that the trefoil knot can be deformed - in this case simply by turning the whole knot around - into an exact copy of itself except that the direction of fluid flow has been reversed. If you can do this to a knot, then the knot is invertible .
This figure of eight knot is also invertible. Can you see why?
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Most small knots are invertible. In this table of small knots with up to 8 crossings only one is listed as not invertible. We're not sure about the knot listed there as {8,10} - you might be able to help us decide on that one!
You can use the symmetry of many of the drawings to convince yourself that the knots are invertible, but sometimes the drawing doesn't help and you'll have to get a piece of string and make the knot for yourself before you can see it.
Here are some of the more interesting cases to try. Be careful, the first one is trickier than it looks!
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Getting Started
A good start is to make the knots you can see on the problem page.
The thicker the string the easier it is to see the 'shape' of the knot.
Rope is very good, and a tube length can be joined to make a complete circuit.
This is good new mathematics so be prepared to invest a bit of time getting equipped so you can enjoy exploring these ideas.Student Solutions
No solution yet with a good explanation
Teachers' Resources
Work with this topic is most productive when the theme can be given some time.
Because it is not traditional school mathematics it offers a good opportunity for students to explore something new and experience the emergence of structure for themselves.
Having some good weight rope is helpful and the use of tubing
allows easy closure to present a closed path in three dimensional
space.
Allow plenty of time for discussion :
How do we begin to talk about something which for most of us is just a tangle of string or a tied knot we do so automatically, like a shoe lace or a reef knot, that we hardly think about it ?
A knot collection might make a stimulating talking point for students in a classroom.