Knot theory originated in an idea of Lord Kelvin's, that atoms were knots of swirling vortices in the æther, and that an understanding and classification of all possible knots would explain why atoms absorb and emit light at only the discrete wavelengths that they do (i.e. explain what we now understand to depend on quantum energy levels). The vortex theory died, but knot theory has grown into a subject with wide and often unexpected applications, for example to theories of particle physics, DNA replication and recombination, and to areas of statistical mechanics.
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A useful way to visualise knots and the allowed moves on them is to project the knot onto a plane - think of the knot casting a shadow on the wall. Now we can draw and manipulate pictures, instead of having to think in 3D. However, there is one more thing we must do - at each crossing we must indicate which section is "over" and which is "under". This is to prevent us from pushing one piece of string through another, which is against the rules. To avoid ambiguity, we must avoid having three arcs cross at the same crossing and also having two arcs meet without actually crossing (we would say that the knot is in general position with respect to the plane). Fortunately a small perturbation in either the original knot or the position of the plane is all that is needed to ensure this.
{Note: This will be a lot easier to follow when there are some diagrams here!}
I. Twist and untwist in either direction.
II. Move one loop completely over another.
III. Move a string completely over or under a crossing.
Reidemeister was the first to mathematically demonstrate that knots really exist - that is, that there really are knots that are not equivalent to the unknot. He did this by inventing the first knot invariant, demonstrating a property of a knot diagram which is not changed when we apply any of the Reidemeister moves.