Differential Geometry and Topology of Curves 2024, 42: Linked Curves & Knots

Two disjoint closed space curves are not linked, if they can be deformed continuously into curves situated inside two disjoint balls. Otherwise they are linked. From their position vectors, the link coefficient can be constructed through the Gaussian integral

For non-linked curves, then I = 0. There are some linked curves with a link coefficient of 0. An example would be two curves, of which one is linked with itself, but in a different winding direction than the way that the other curve links with the resulting loop. Regular curves of class C³ are boundaries of a plane strip iff

Knots are closed space curves without self-intersections. Polygonal knots are the union of a finite number of edges. A knot is tame (as opposed to wild) if there is a homeomorphism of the space onto itself, mapping the knot onto a polygonal one. C¹-regular knots are tame. For any tame knots, the fundamental group E³\γ is the group of γ, a particular case of the Poincare group. It takes a point O outside γ and a set of all closed oriented space curves passing through O, which are disjoint of γ. For these curves, consider O the starting and end point. They are all images of [0, 1] under a map g so that g(0) - g(1) = O. Any two of these curves are equivalent if one can't be continuously deformed into the other without intersecting γ. The resulting equivalence classes give rise to (usually non-commutative) groups G(γ). For the group generators, project γ into a "horizontal" (the name is convention, the plane is arbitrary) plane. The projection has only double self-intersections. The arc of the projection is shown without a break at any self-intersection point P for overpassing arcs, and with a break for underpassing ones. This makes the resulting curve unique to the projection. The oriented knot can then be decomposed into oriented connected arcs. For any arc, a trivial knot can be constructed that passes through O such that it can't be continuously deformed into O without intersecting γ and such that the orientations of γ and the arc are connected via the screw rule. Then, all curves in G(γ) are equivalent to products of a product of such arcs. The equivalent classes of these arcs can be viewed as generators of the knot group.

A knot is trivial iff its knot group is isomorphic to the additive group. For two knots in the 3-sphere their classifying groups are isomorphic, if the knots are equivalent. The classifying groups are defined as CG(γ) = G(l+)*G(l-), where l are complementary windings. Knots are always boundaries of an oriented surface.