## 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.

To proceed to the discussion of showing Knot Inequivalence click
**here**.

*Charilaos Aneziris, charilaos_aneziris@standardandpoors.com*

**Copyright 1995**

### COPYRIGHT STATEMENT

Educational institutions are encouraged to reproduce and
distribute these materials for educational use free of
charge as long as credit and notification are provided.
For any other purpose except educational, such as
commercial etc, use of these materials is prohibited without
prior written permission.