One of the irritating things about writing mathematics is that one has to be accurate in what one says, but one also WANTS to speak comprehensibly. Here is what I am trying to say: “for a certain subclass X of wild knots, to each equivalence class of knots of class X there corresponds a sequence of equivalences classes of tamely embedded solid tori.” This is accurate but obscures what I am trying to say: sequences of solid tori are a knot invariant for knots of type X.”
(If you are wondering what I am talking about: a tame solid torus can be thought of as a possibly knotted solid bagel in space. A smooth knot is an image of a differentiable, one to one map of the unit circle into 3 space, and a wild knot is an image of a continuous map of the unit circle into 3 space which cannot be deformed (by a continuous, one to one function of 3-space to itself) into a smooth knot.
This is an example of a wild knot; one can define a tangent vector to every point of this knot EXECPT for one exceptional point. It is that point that makes the knot “wild”.
The knots I am studying are wild at ALL of their points.
Note: if you wondered “why is he being so stilted in his definition of “wild knot””, here is why: consider the following image of a circle: join the graph of to the semi-circle from the graph: . This forms the continuous, non-differentiable, one to one image of a circle. But it is NOT wild; it can be shown that this knot is actually equivalent to a differentiable knot (the “unknot” actually).
This is an excellent example of precision conflicting with ease of understanding.