Probably the most important thing proved in knot theory is that knots exist. In the early days of knot theory, people tried to distinguish knots by tediously bending and twisting them. If one knot turned into another, they were the same. If no method of turning one knot into another was found, the knots might be different. Unfortunately, they could not use trial and error to prove that there was such a thing as a knot, for someone can undertake the job of untying a knot countless many ways and totally overlook the solution.
In order to prove that knots exist, there must be some property of a knot, known as an isotopy invariant that remains constant regardless of the form. An isotopy invariant could be determined for a knot and an unknot. If the properties do not match, then it is proven that the knot cannot become the unknot and therefore knots do exist.
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