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## 'The Invertible Trefoil' printed from http://nrich.maths.org/

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.

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?

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!