Knots, Trefoil Knot
The Trefoil Knot
The trefoil knot is the simplest nontrivial knot.
Start with a triangle and add a circular arc to each side.
Thus the shape has three leaves, or three foils,
as suggested by the latin ancestry of the word "trefoil".
Yet this is merely a 2 dimensional shape; we need to use 3 dimensions to turn it into a knot.
Start at the circular arc below the base of the triangle and move counterclockwise, to the right and up,
until you reach the lower right corner of the triangle.
Pass over that line and move up the right side of the triangle.
When you reach the apex, pass under that line and procede around the left circular arc.
Pass over the next line, follow the base of the triangle,
pass under, follow the right circular arc up,
pass over, run down the left side of the triangle, pass under,
and return to start.
The crossings run over, under, over, under, over, and under.
Try making this shape with a piece of string - fusing the two ends together of course.
Some experimentation will convince you this is a true knot;
you cannot move it about and make a circle.
Other n-gons
Try the same thing with a square.
Place four circular arcs on the four sides, and let the string run over, under, over, under -
in alternation.
The result is two separate, intertwined strings.
This is equivalent to a figure 8
with a loop passing through both holes.
Characterizing knots is hard enough
without generalizing the problem to multiple intertwined loops,
so I'm going to set this case aside for now.
If n is odd, a regular n-gon, with circular arcs, and alternating crossings,
defines a knot.
Proving these knots are nontrivial, and distinct from one another,
is a task that remains before us.
Return to the trefoil knot, and follow two paths simultaneously.
Start at a point in the lower arc,
and a point in the base of the triangle,
and follow both paths counterclockwise.
The circular arc moves up to pass over the base of the triangle, which moves down out of the way.
In other words, the two threadss twist around each other.
As they move from the lower left corner of the triangle to its apex, they twist another 180 degrees, to change the crossing.
The next leg of the journey introduces another twist.
finally they return to start, 180 degrees out of phase.
With this in mind, the trefoil knot can be drawn on the torus.
It is a curve of constant pitch.
The string runs around the doughnut twice, while spiraling around the tube three times.
The ratio is 3 to 2, or 3/2 if you prefer.
The trefoil knot is a 3/2 torus knot.
Using the same reasoning, the knot based on the pentagon (with circular arcs) is a 5/2 torus knot.
It spins around the tube 5 times while circumnavigating the torus twice.
Equivalently, once around the torus introduces a twist of 5/2.
If the fraction is not in lowest terms, having a gcd of g,
then completing all the prescribed revolutions and rotations actually runs through the knot g times.
We may as well assume the fraction is in lowest terms.
Alternatively, you might interpret an unreduced fraction as g copies of the same knot intertwined.
This was the case with the augmented 4/2 square described above.
Again, I'm going to pass on multiple knots,
so unless otherwise stated, torus knots will be in lowest terms.
The n/1 knot is trivial.
Let c be the circle that runs through the center of the torus, and homotope the spiral inward, onto the circle.
The 1/n knot is also trivial, though the homotopy is not as straight forward.
For large n, some of the spirals can be seen close together, running around the top of the doughnut.
this resembles the burner on an electric stove.
Let q be a point on the path at the bottom of the torus,
and stretch q into an arc xy that curves upward around the bottom of the tube.
This crams the spirals together.
soon all the spirals are on top, with x and y delimiting the n spirals,
like the two leads that feed the burner on the stove.
The arc xy goes almost all the way around the tube, passing underneath.
Now - discard the torus and leave the knot in 3 space.
Let y run all the way around the spiral, dragging the arc xy along.
After y has traced around all n spirals, it is back to x, and the result is a circle in space.
This is a trivial knot.