Since the website where I originally hosted this page crashed and sent my old pictures into oblivion, I'll have to use words to describe the tangle: Think of a collection of macaroni pieces strung together in a necklace so that the circular ends of each piece are flush against the circular ends of both neighbors, and each piece is exactly 90 degrees of arc (a quarter of a circle, or a right angled elbow). In real life, the pieces are brightly colored PVC, and so the assembly looks like a cross between a biochemist's protein folding model and a chew toy.
The junction between the ends of two neighboring pieces is a
joint which can swivel a full 360 degrees. But the system is
highly constrained - as you twist one joint the others also move.
So if you imagine making marks across each of the joint seams,
you can describe the position of the toy as a collection of n
angle measurements, where n is the number of joints (which is
also the number of segments). Another way to say this is that
the space of toy positions is a subspace of the n-torus:
There are further questions we can ask if we take into account the "chubbiness" of the pieces of the toy (the ratio of the cross sectional radius of a piece to the radius of the loop of material from which the piece appears to have been cut), and if we allow the pieces to be something other than 90 degrees of arc. But the simplest case, in which the pieces are allowed to pass through each other, is tough enough.
Another way to think of the configuration space (of the simplest case) is as a real algebraic variety : we want the space of ordered n-tuples of unit vectors v_1, v_2, ..., v_n in Euclidean 3-space so that the sum is zero, v_i is perpendicilar to v_{1+1} for i = 1, ... n-1, and v_n is perpendicular to v_1. But this doesn't help a whole lot, because systems of quadratic equations are difficult to solve.
Tangles were at the height of their popularity in 1989 and 1990, when a Washington State company called Think! manufactured them, and a number of different toy chains distributed them (I got most of mine at the Imaginarium in Los Angleles). At the time, they were available in 4 sizes and lots of different colors. The largest one I've ever seen had segments which were about the size of a small banana, but it sure would be cool to have a behemoth model machined out of stainless steel (with bearings to make the action really smooth). This would take the "flex" out of the joints, and make it easier to "feel" the geometry of the configuration space.
Building a virtual tangle would be pretty cool, too. But I think you have to solve the problem of describing the configuration space for n segments before you can build one. If you're fantasizing about "searching all of T^n for points in the subspace", be warned that brute force won't yield much insight per hour of supercomputer time wasted.