72 L.H. Kauffman and S. Lambropoulou
A two-tangle is an embedding of two arcs (homeomorphic to the interval [0,1])
and circles into a three-dimensional ballB^3 standardly embedded in Euclidean
three-spaceS^3 , such that the endpoints of the arcs go to a specific set of
four points on the surface of the ball, so that the circles and the interiors
of the arcs are embedded in the interior of the ball. The left-hand side of
Fig. 5.1 illustrates a two-tangle. Finally, a two-tangle isorientedif we assign
orientations to each arc and each circle. Without loss of generality, the four
endpoints of a two-tangle can be arranged on a great circle on the boundary
of the ball. One can then define adiagramof a two-tangle to be a regular
projection of the tangle on the plane of this great circle. In illustrations we
may replace this circle by a box.
The simplest possible two-tangles comprise two unlinked arcs, either hor-
izontal or vertical. These are thetrivial tangles, denoted [0] and [∞] tangles,
respectively, see Fig. 5.2.
Definition 1 A two-tangle isrationalif it can be obtained by applying a
finite number of consecutive twists of neighbouring endpoints to the elemen-
tary tangles [0] or [∞].
The simplest rational tangles are the [0], the [∞], the [+1] and the [−1]
tangles, as illustrated in Fig. 5.3, while the next simplest ones are:
(i) Theinteger tangles, denoted by [n],made ofnhorizontal twists,n∈Z.
(ii) Thevertical tangles, denoted by 1/[n], made ofnvertical twists,n∈Z.
These are the inverses of the integer tangles, see Fig. 5.3. This terminology
will be clear soon.
3-22,
Fig. 5.1.A two-tangle and a rational tangle[0] [ ]
,
Fig. 5.2.The trivial tangles [0] and [∞]