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In knot theory, the crossing number is an example of a knot invariant. A knot's crossing number is simply the lowest number of crossings of any diagram of the knot.
By way of example, the unknot has crossing number zero, the trefoil knot three and the figure-eight knot four. There are no other knots with a crossing number this low and just two knots have crossing number 5, but the number of knots with a particular crossing number increases rapidly as we go higher.
Tables of prime knots are traditionally indexed by crossing number, with a subscript to indicate which particular knot out of those with this many crossings is meant (this sub-ordering is not based on anything in particular, except that torus knots are listed first). The listing goes 31 (the trefoil knot), 41 (the figure-eight knot), 51, 52, 61, etc. This order has not changed significantly since P. G. Tait published a tabulation of knots in 1877.
The crossing number cr(G) of a graph G is the lowest number of crossings of a planar diagram of the graph G. For instance, a graph is planar if and only if its crossing number is zero.
The very useful crossing number inequality, discovered independently by Ajtai, Chvatal, Newborn, and Szemerédi [1] and by Leighton [2], asserts that if a graph G (undirected, with no loops or multiple edges) with n vertices and e edges has many edges, in the sense that if
then we have
The constant 33.75 is the best known to date, and is due to Pach and Tóth [3]; the constant 7.5 can be lowered to 4, but at the expense of replacing 33.75 with the worse constant of 64.
The motivation of Leighton in studying crossing numbers was for applications to VLSI design in theoretical computer science. Later, Székely [4] also realized that this inequality yielded very simple proofs of some important theorems in incidence geometry, such as Beck's theoremThere are two (completely different) theorems in mathematics (by two different mathematicians) going under the name of Beck's theorem . Beck's theorem in category theory Beck's monadicity theorem asserts that a functor : is monadic if and only if # U has and the Szemerédi-Trotter theoremIn mathematics, the Szemeredi-Trotter theorem is a result in the fields of combinatorial geometry and irregularities of distribution. It asserts that given n points and m lines in the plane, the number of incidences (i. the number of point-line pairs, suc.
We first give a preliminary estimate: for any graph G with n vertices and e edges, we have
To prove this, consider a diagram of G which has exactly cr(G) crossings. Each of these crossings can be removed by removing an edge from G. Thus we can find a graph with at least edges and n vertices with no crossings, and is thus a planar graph. But from Euler's formula we must then have , and the claim follows. (In fact we have for n≥ 3).
To obtain the actual crossing number inequality, we now use a probabilistic argument . We let 0 < p < 1 be a probabilityThe word probability derives from the Latin probare (to prove, or to test). Informally, probable is one of several words applied to uncertain events or knowledge, being more or less interchangeable with likely risky hazardous uncertain and doubtful depend parameter to be chosen later, and construct a random subgraphIn mathematics, a random graph is a graph that is generated by some random process. The theory of random graphs lies at the intersection between graph theory and probability theory, and studies the properties of typical random graphs. Random graph models H of G by allowing each vertex of G to lie in H independently with probability p, and allowing an edge of G to lie in H if and only if its two vertices were chosen to lie in H. Let denote the number of edges of H, and let
denote the number of vertices.Now consider a diagram of G with cr(G) crossings. We may assume that any two edges in this diagram with a common vertex are disjoint, otherwise we could interchange the intersecting parts of the two edges and reduce the crossing number by one. Thus every crossing in this diagram involves four distinct vertices of G.
Since H is a subgraph of G, this diagram contains a diagram of H; let denote the number of crossings of this random graph. By the preliminary crossing number inequality, we have
Taking expectations we obtain
Since each of the n vertices in G had a probability p of being in H, we have . Similarly, since each of the edges in G has a probability of remaining in H (since both endpoints need to stay in H), then . Finally, every crossing in the diagram of G has a probability of remaining in H, since every crossing involves four vertices, and so . Thus we have
If we now set p to equal 4n/e (which is less than one, since we assume that e is greater than 4n), we obtain after some algebra
A slight refinement of this argument allows one to replace 64 by 33.75 when e is greater than 7.5 n; see [3].