Complementary Regions of Knot and Link Diagrams

Abstract

Abstract. An increasing sequence of integers is said to be universal for knots and links if every knot and link has a projection to the sphere such that the number of edges of each complementary face of the projection comes from the given sequence. In this paper, it is proved that the following infinite sequences are all universal for knots and links: (3, 5, 7, . . . ), (2, n, n + 1, n + 2, . . . ) for all n ≥ 3 and (3, n, n + 1, n + 2, . . . ) for all n ≥ 4. Moreover, the following finite sequences are also universal for knots and links: (3, 4, 5) and (2, 4, 5). It is also shown that every knot has a projection with exactly two odd-sided faces, which can be taken to be triangles, and every link of n components has a projection with at most n odd-sided faces if n is even and n + 1 odd-sided faces if n is odd.

Universal knot diagrams

“…The following result in this direction was established by von Hotz [VH60] and by Ozawa [Oza07]. See also the more recent works, [AST11], [Owa18] and [BM18]. Theorem 1.2 follows from a simple construction that starts with an arbitrary knot diagram and produces a meander diagram of the same knot.…”

Meander diagrams of knots and spatial graphs: Proofs of generalized Jablan–Radović conjectures

“…2; each of newly emerged crossings involves arcs representing distinct edges, so that a finite number of such moves yields a semi-meander diagram. For the case of knots, the idea of obtaining semi-meander diagrams in this way appears in [Oza07] (see also [AST11]). 2.…”

“…In 2007, an elegant idea of a much shorter proof for the statement of Conjecture (C1) has appeared in the preprint [Oza07] of M. Ozawa. A variation of Ozawa's method is described in detail in [AST11]. In addition to the above, the existence of a semi-meander diagram for each knot becomes an obvious corollary of a much more general result about spatial graphs due to R. Shinjo [Shi05], 2005, if we treat a knot as a spatial graph with two vertices and two edges (below, this is explained in more detail).…”

Axis systems of link projections

“…For a region of a link projection, we call it an n-gon if the region has n edges on the boundary. There are many studies about n-gons in knot theory (see, for example, [1,2,6]). Now we define an axis; Choose a start point and an orientation at an edge or a crossing as depicted in Fig.…”