Knots, An Introduction
Introduction
Your shoelace might be tied in a knot, but this isn't a mathematical knot,
because the ends are free.
You can always pull the ends through in a way that unties the knot, and returns the shoelace to a straight piece of string.
A topological knot has no free ends.
If you like, the two ends are fused together.
It is basically a loop in 3 space.
The loop must not touch itself.
In other words, the map from the circle (domain)
onto the loop in 3 space (range) is bijective.
And of course the map is continuous, as is required by any loop.
To keep things simple, the knot must also be piecewise smooth.
Let's not try to manage infinitely fuzzy fractals.
At some level, the knot should look like a string,
perhaps a very thin string,
that runs through 3 space, with its ends joined together.
The string may weave under and over and around itself; and that's when the knot becomes interesting.
Free Ends
Assume for a moment that the knot has free ends, like your shoelace.
In other words, the knot is a path, not a loop.
And it exists in any number of dimensions.
Remember that the path is piecewise smooth,
hence arc length is well defined.
Find a point q on the path where it is continuously differentiable.
Let p be the preimage of q, i.e. the point on the interval [0,1] that maps to q.
Contract the interval linearly towards p,
and at the same time, contract the path towards q.
The path shrinks along its length,
as though the two aglets were eating string and moving towards each other.
This is a well behaved homotopy.
Stop when the path is within δ of q,
such that the tangent vector is always within ε of the tangent at q.
In other words, the path is almost a straight line.
A second homotopy straightens out the last little bit of curve in the path,
and the result is a line segment.
Finally, stretch this segment to unit length.
Therefore, every open ended knot is equivalent to [0,1].
I mentioned this earlier;
you can always untie your shoelace.
Distinct Knots
With the open ended string put to bed,
let's return to the true knot, a closed non-intersecting loop in 3 space.
Two such knots are topologically distinct if one cannot be transformed into the other through a homotopy.
We have to be a bit careful here.
Any two loops become homotopic in real space.
However, the knot is suppose to represent a string that cannot crunch down to a point or pass through itself.
Therefore, the homotopy must always maintain a bijective map from start to finish.
As the knot moves about in space, at no time does it touch itself.
It can wiggle about, or stretch, or shrink,
but it cannot break, crunch down to a point, or pass through itself.
If one knot continuously transforms into another,
while obeying these restrictions,
then they are essentially the same knot.
Trivial Knot
The trivial knot is the circle.
This is of course the same as an ellipse, or a square, or a whole host of shapes floating in 3 space
that deform continuously into a circle.