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.
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, firstname.lastname@example.org
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