A knot is a simple closed curve (homeomorphic image of S(1)) in Euclidean 3-space E(3). Two knots are called equivalent when there is an orientation-preserving homeomorphism of E(3) onto itself sending one knot to the other.
Schoenflies proved in 1908 that any homeomorphism from a simple closed curve in the plane E(2) onto the unit circle S(1) can be extended to a homeomorphism of the plane onto itself. Similar things do not hold in higher dimensions.
For example, there exist wild embeddings of simple arcs into E(3): homeomorphic images of the unit interval such that the complement is not simply connected. Thus, one usually restricts knots to be tamely embedded, e.g., as a simple closed polygonal curve, and we'll do so as well.
Two more examples of wild embeddings:
Example (Alexander's horned sphere) A homeomorphic image of the sphere S(2) in E(3) such that the complement of the image is not simply connected.
Example (Antoine's necklace) A homeomorphic image of the Cantor set (which is compact and totally disconnected) in E(3) such that the complement of the image is not simply connected.
(The pictures here were taken from Hocking & Young, Topology, pp. 176-177.)
Given a knot in E(3), we project it from a point in general position into E(2), so that the resulting curve never passes three times through the same point, and indicate for each crossing whether it is an over- or undercrossing. The resulting diagram suffices to retrieve the knot up to equivalence.
Now equivalence between knots can be translated into equivalence between diagrams. This was done by Reidemeister, who showed that two diagrams represent the same knot if and only if one is obtained from the other by a sequence of Reidemeister moves:
in each of the three types of move, we may replace the upper picture by the lower, or vice versa; type I also has a mirror image, type I'.
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