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Spherical knot is a knot made by one or more ropes "travelling" on the surface of a sphere.
One rope makes a stopper, two ropes make a bend ( 2-rope link ).
The word "sphere" is to be meant with its topological, rather than its geometrical sense. Any 2-D closed surface, topologically equivalent to the sphere, is, in this sense, "spherical". The elastic surface of a balloon, independently of how much we inflate or we deformed it, locally, by squeezing it, is and remains "spherical", as long as we do not tear it apart.
All the segments of the ropes cross each other, in pairs, at some points, the "crossing points ". So we should imagine that the crossing points are "
points" distributed on the surface of the sphere, and there are two "
lines" ( two segments of the one rope, in the case of a stopper, or the two ropes, in the case of a bend ) that cross each other at those "points", going "over" or "under".
Of course, any knot can be "loosened", and then "flattened", so it will look like the 2-D representations of knots used by the mathematicians. The advantage of a spherical knot is twofold :
First, the crossing points of a spherical knot can be arranged in such a way that their number is minimal. On the contrary, most of the times, on a loose and flattened knot this will not happen.
How can we achieve this minimum number of crossing points ? By just
copying the diagram of the representation of the corresponding mathematical knot or link shown in a table of knots or links ( which, by definition, will have the
minimum number of crossing points ) and then
pasting / "embedding" it on the surface of a sphere. In other words, it is like we draw such a 2-D representation of our stopper or our bend on a piece of a rubber sheet, and then we wrap a sphere with it, so we cover it completely. A knot that looks like this is spherical, and has the minimum number of crossing points on its surface.
Second, in a spherical knot all the distances between all the crossing points remain small - because a sphere with a number, say n, of crossing points on its surface, is more compact, spatially, than a planar area with the same number and density of crossing points. By wrapping the sphere with the planar 2_D drawing, the points that use to be on the perimeter of it will come closer together, on the 'rear" surface of the sphere.
Those two advantages are especially helpful in the case we want to "inflate" the spherical knot, in order to be able to reduce the density of the crossing points on its surface, so the lines that go "over" or "under" those points are more clearly visible. Also, when we have a spherical knot, we can rotate it and look at it from the angle which will show the interweaving of the segments of the rope(s) in a more comprehensible way.
As an example of a relatively simple, most symmetric spherical knot, the interested reader may try his hand and tie a bend based on a quasi-regular polyhedron ( an
Archimedean solid ) with 12 vertices, the
cuboctahedron ( see the attached picture ). The blue line represents the continuous path of the one rope, of the first link, on the surface of the one hemisphere, and the red line the continuous path of the other rope, of the second link, on the surface of the other hemisphere. Starting from the "closed" knot, we can "cut" the red and the blue slings at one red and at one blue edge, respectably, and get two pairs of free ends, i.e, get an
end-to-ender knot, a bend, like the ones we use. By altering the "over" / "under" relations of the pairs of lines at each crossing point, we can get different bends.
If we want the most symmetric such knot, the "cuts" should be on symmetrically located edges, and the "over" / "under" relations would form two interlinked mirror symmetric
trefoil knots, i.e., two trefoil knots of opposite handedness / chirality - which, when we will "cut" them, they will be transformed in one "cut" left-handed trefoil knot = an overhand knot, and one "cut" right-handed trefoil knot = an underhand knot. The corresponding polyhedron can be seen, animated, in :
http://upload.wikimedia.org/wikipedia/commons/a/ab/Cuboctahedron.gif
