Knots and their polynomials

Knots as topological objects

Mathematicians have studied knots for the last 150 years or so, or ever since Topology started to cristallize as a mathematical discipline. One classical problem is to show that the left-handed and right-handed knots really are different. To keep themselves honest, mathematicians stick the free ends of each knot together, to form a loop which holds the knot captive. When overhand knots are made into loops, the resulting configuration has a three-fold symmetry, like the outline of three leaves. Hence the names Left-handed Trefoil and Right-handed Trefoil for the closed form of these knots.

Left trefoil Right trefoil
Left and right-handed trefoils.

Topologists consider two knots to be the same if one can be deformed to match the other. The statement ``Left and right-handed trefoil are different'' means to a topologist that no matter how the string is stretched or twisted in three-dimensional space, there is no way to change one into the other without tearing the string.

How do you prove mathematically that something can't be done? The usual way in topology is to figure out some quantity that can be calculated from a configuration, and such that when the configuration is changed following the rules (here: without tearing the string) that quantity does not change. Such a quantity is called an invariant. If you can find an invariant which gives different values for the left and right-hand trefoil, that will prove that one cannot be deformed to the other.

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