id: 06331439
dt: j
an: 2014e.00629
au: Adams, Colin; MacNaughton, NĂ¶el; Sia, Charmaine
ti: From doodles to diagrams to knots.
so: Math. Mag. 86, No. 2, 83-96 (2013).
py: 2013
pu: Mathematical Association of America (MAA), Washington, DC
la: EN
cc: H75
ut: closed curves; knots: $(2,n)$-curves
ci:
li: doi:10.4169/math.mag.86.2.83
ab: Summary: What closed curves can be drawn in the plane such that they cut
the plane into complementary regions that are $n$-gons, including the
outer region, where $n$ is allowed to take some finite number of
values? A curve is an $(a_1, a_2,\ldots,a_n)$-curve if the number of
edges for its complementary regions all lie in $(a_1,a_2,\ldots,a_n)$.
We show that there are infinitely many curves for $(2, n)$, where $n$
is any odd integer greater than 3, and for $(3, n)$, for any $n > 3$
relatively prime to 3. We also consider the implications for knot
theory, showing that every knot has a $(3, 4, 5)$-diagram. We ask what
values of $(a_1,a_2,\ldots,a_n)$ will generate diagrams for every knot.
rv: