id: 05999799
dt: j
an: 05999799
au: Adams, Colin; Shinjo, Reiko; Tanaka, Kokoro
ti: Complementary regions of knot and link diagrams.
so: Ann. Comb. 15, No. 4, 549-563 (2011).
py: 2011
pu: Birkhäuser Verlag (Springer), Basel
la: EN
cc: 
ut: knot diagram; complementary region; 4-valent graph
ci: 
li: doi:10.1007/s00026-011-0109-2
ab: 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.
rv: Iain Moffatt (Mobile, AL)