There's an interesting feature of many knots called "fibring", and such knots are called "fibred knots". A knot can be thought of as living in R^3 but it can be more useful to think of it as living in the 3-sphere S^3 (which is topologically just R^3 with a point added at infinity). To distinguish knots — since all knots are topologically circles — people have studied not the knot itself but what becomes of S^3 after the knot is removed. S^3 - K is called the knot complement. For many knots K in S^3, it turns out that there is a lovely surjective map π : S^3 - K —> S^1 to the circle, such that the inverse image of each point of S^1 is always a certain M (with boundary) in a continuous manner. If we removed one of these surfaces, say π^(-1)(1), what remained of S^3 - K would be topologically M x (0, 1), the product of the surface with an open interval. If we like we can glue on the missing end-surfaces to get M x [0, 1]. Then knot complement is completely determined by the way the end-surfaces are glued back together: S^3 - K = M x [0,1] / (x,1) ~ (h(x),0) where h is some self-homeomorphism h : M —> M of M. The trefoil can be thought of a the intersection of the zero-locus of the polynomial P : C^2 —> C (C = complex numbers) defined by P(z,w) = z^3 + w^2 with a small 3-sphere S^3 about the origin (0,0) of C^2. Then on S^3 - K, that polynomial is always non-zero. So we can define a mapping π : S^3 - K —> S^1 via π(z,w) = P(z,w) / |P(z,w)| and that defines the fibring as above. For the trefoil, it turns out that the fibre M is just a torus with a disk removed. Even the unknot (a geometric circle) has a fibring: The fibre is just a disk. But some knots have no such fibring. —Dan