Tangle (mathematics)
 

a tangle is generally one of two related concepts:

In John Conway's definition, an n-tangle is a proper embedding of
the disjoint union of n arcs into a 3-ball;
the embedding must send the endpoints of the arcs to 2n
marked points on the ball's boundary.

In link theory, a tangle is an embedding of n arcs and m circles
into R^2 X [0,1] – the difference from the previous definition
is that it includes circles as well as arcs,
and partitions the boundary into two (isomorphic) pieces,
which is algebraically more convenient –
it allows one to add tangles by stacking them, for instance.

Conway's sense of tangles

Two n-tangles are considered equivalent if there is an ambient isotopy of one tangle to
the other keeping the boundary of the 3-ball fixed.

Tangle theory can be considered analogous to knot theory except instead of closed loops
we use strings whose ends are nailed down. See also braid theory.