| In the set of natural
numbers N with multiplication inverse elements do not exist. The same holds
for KLs: inverse KLs do not exist. One KL cannot cancel out another. That
is, for a given KL there is no KL that, composed with the first, gives
the unknot (unlink). A beautiful proof of the imposibility of knot cancellation
is given by J. Conway (Gardner, 1986).
One way that composition of knots differs from multiplication of natural numbers is that there is more then one way to form the composition of two knots. There is a choice of where we remove an arc from the outside of each projection in order to get free strands and connect two knots. From the same pair of knots K1 and K2 it is often possible to construct two different composite knots.
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