College Math Teaching

February 13, 2013

Writing Math: being comprehensible and accurate at the same time….ARGGGHHH!!!

Filed under: advanced mathematics, calculus, editorial, research, topology — collegemathteaching @ 7:47 pm

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.

wildKnot

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 f(x) = xsin(\frac{1}{x}) to the semi-circle from the graph: g(x) = -\sqrt{\frac{1}{4} - (x - \frac{1}{2})^2}. 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.

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