The Arithmetic of Knots

Here is how to add two knots.

Pull out a small section from each and then join them up.

Joining one of these knots onto a different part of the other knot might seem to change their sum. We can show that it does not by manipulating one knot around the other. From this we can make a general rule about the addition of knots:

K + L = L + K.
This is called commutativity.

What other rules can we find?

The picture on the left shows that if we add the unknot O to K we simply get K.
Another rule arises from adding three knots together. The rule is now simply

K + (L + M) = (K + L) + M.
This rule is called associativity.


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© Mathematics and Knots, U.C.N.W.,Bangor, 1996 - 2002
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