## Showing Knot Equivalence

### Regular Projections

While knots are embedded in three dimensions, one usually studies their two-dimensional projections (projections on a plane or a two-sphere). The projections that are usually considered are the so-called regular ones, which satisfy the following properties.
• No more than two points of a knot are allowed to be projected on the same point of the two-dimensional surface.
• Let a knot defined through f, and f(s) a point of the knot. The tangent at s, f'(s)=df/ds, is not allowed to be perpendicular to the projective surface.
• Let f(s) and f(r) two points of a knot, and f'(s) and f'(r) the tangents at these two points. The differences f(s)-f(r) and f'(s)-f'(r) are not allowed to be simultaneously perpendicular to the projective surface.
• At each crossing one distinguishes between the overcrossing and the undercrossing segment.

All knots possess regular projections; in fact most of the projections do satisfy the properties above, since an infinitesimal change of a non-regular projection gives a regular one. For examples of projections that are not regular, click here.

### The Reidemeister Moves

In the 1920's Kurt Reidemeister proved the following theorem.

Any two regular projections of ambient isotopic knots are related through a finite series of moves.
There are just three kind of such equivalence moves, which are called the Reidemeister moves. If you want to have a look at the Reidemeister moves, click here.

While it may seem obvious that performing a Reidemeister move leads to an ambient isotopic knot, the important fact is the sufficiency of these moves: if two knot projections cannot be connected by a series of such moves, they definitely belong to inequivalent knots.

The theorem however does not provide any upper boundary for the number of moves that may be necessary; therefore failure to connect two projections after some number of moves does not establish inequivalence; it may even be necessary in order to connect equivalent knots possessing n and m crossings, to pass through knots with more than max(m,n) crossings.