Definition   An n-twist is applied to a tangle T by choosing an adjacent pair of ends and twisting them around each other producing n crossings, where n is an integer. eg a North 3-twist:
Definition   A rational tangle is the
result of applying a sequence of twists (various directions) starting
with 0 or .
From the symmetry in the definition it is clear that the dihedral
group of 8 elements D8=< r,y | ry =
yr3, r4 = 1 = y2> acts on the set of (isotopy classes of)
rational tangles, where r acts by rotation about the x-axis
through 900 and y is rotation through 1800 about
the y axis. Let x = r2. Let also z = xy then z acts by
rotation through 1800 about the z-axis.
The following is perhaps a little surprising.
Proposition  x, y, z operate trivially on rational tangles.
Proof  Since z = xy it
will be sufficient to prove that x and z operate trivially. Clearly
for one n-twist applied to 0 or the result holds. The proof now proceeds by
induction on the number of twists applied.
Case i)   If the twist is east or west then y
operates trivially, since we have yR = R by induction:
To see that z
operates trivially operate on zR by yn thus transferring
crossings from west to east say and use ynzR = R by
induction:
 
Case ii)   If the n-twist is north or south then z operates trivially and y can be seen to operate trivially much as in case i).
Remark  The operation on part of a knot or link
given by:
We now define a basic tangle (corresponds to rational
tangle in the lectures) to be a rational tangle obtained from 0 or
by using only
south and east twists. By using flypes and the the proposition we can
convert west twists into east twists and north twists into south
twists. Hence we have:
Proposition  Every
rational tangle is isotopic to a basic tangle.