I've defined rational tangles but I should set that in a wider
context. A knot is essentially a single closed loop embedded in
3D space and a link is a non-intersecting union of a bunch of
these. Any tangle, knot or link can be projected down to 2D
space so that you get a finite collection of 'over' or 'under'
crossings connected by arcs. You might have to jiggle things
around a bit to ensure that you don't get any degeneracies like
two separate parts of a knot being projected to the same segment
of curve in 2D, but this is always possible. Here's an example
of such a 2D projection, a knot diagram, for the photograph I
posted last week: Roughly speaking, two knots or links or
tangles are equivalent if you can slide the strings about to get
from one to the other without passing one string through
another. In the case of rational tangles we have the extra
constraint that the free ends mustn't move and must always
remain 'at infinity' so you can't pass loops over the end. Given
two diagrams for rational tangles,