<B> Showing Knot Equivalence

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

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