Knots, Links and Tangles

We start with some terminology from differential topology [1]. Let  be a circle and  ≥ 2 be an integer. An immersion  :  → R is a smooth function whose derivative never vanishes. An embedding  :  → R is an immersion that is oneto-one. It follows that () is a manifold but () need not be ( is only locally one-to-one, so consider the map that twists  into a figure eight). A knot is a smoothly embedded circle in R; hence a knot is a closed spatial curve with no self-intersections. Two knots  and  are equivalent if there is a homeomorphism R → R taking  onto . This implies that the complements R −  and R − are homeomorphic as well. A link is a compact smooth 1-dimensional submanifold of R. The connected components of a link are disjoint knots, often with intricate intertwinings. Two links  and  are equivalent if, likewise, there is a homeomorphism R → R taking  onto  . We can project a knot or a link into the plane in such a way that its only selfintersections are transversal double points. Ambiguity is removed by specifying at each double point which arc passes over and which arc passes under. Over all possible such projections of or , determine one with the minimum number of double points; this defines the crossing number of  or . There is precisely 1 knot with 0 crossings (the circle), 1 knot with 3 crossings (the trefoil), and 1 knot with 4 crossings. Note that, although the left-hand trefoil  is not ambiently isotopic (i.e., deformable) to the right-hand trefoil , a simple reflection about a plane gives  as a homeomorphic image of . Under our definition of equivalence, chiral pairs as such are counted only once. There are precisely 2 knots with 5 crossings, and 5 knots with 6 crossings. In particular, there is no homeomorphism R → R taking the granny knot # onto the square knot #, where # denotes the connected sum of manifolds [2, 3]. (See Figure 1.) Also, there are precisely 8 knots with 7 crossings, and 25 knots with 8 crossings. A link  is splittable if we can embed a plane in R, disjoint from , that separates one or more components of  from other components of . There are precisely 1, 0, 1, 1, 3, 4, 15 nonsplittable links with 0, 1, 2, 3, 4, 5, 6 crossings, respectively.