05999799

j

05999799 Adams, Colin Shinjo, Reiko Tanaka, Kokoro Complementary regions of knot and link diagrams. Ann. Comb. 15, No. 4, 549-563 (2011). 2011 Birkh\"auser Verlag (Springer), Basel EN knot diagram complementary region 4-valent graph

doi:10.1007/s00026-011-0109-2

The faces of a connected reduced link diagram divide the sphere into a collection of $n$-gons. This paper investigates which collections of $n$-gons can arise from link diagrams. In more detail, a strictly increasing sequence of integers $(a_1, a_2, a_3, \dots )$, with $a_1 \geq 2$, is said to be universal for knots and links if every knot and link has a reduced diagram in which each face is an $a_n$-gon for some $a_n$ in the sequence. (Note that not every $a_n$ in the sequence needs to be realized as an $a_n$-gon in the diagram.) It is shown that the following infinite sequences are universal for knots and links: $(3,5,7,9, \dots)$, $(2,n,n+1,n+2,\dots)$ for each $n \geq 3$, and $(3,n,n+1,n+2,\dots)$ for each $n \geq 4$. In addition, it is shown that the finite sequences $(2,4,5)$, and $(3,4,n)$ for $n\geq 5$ are universal for knots and links. Further results on the collection of $n$-gons that can arise from knot and link diagrams are given. For example, it is shown that every knot has a diagram with exactly two $3$-gons with all other faces even; and it is shown that only sequences with a non-trivial common divisor $k$ arise from links with at least $k$ components. The paper concludes with a short list of questions. Iain Moffatt (Mobile, AL)