caa a22 4500 445855738 CHVBK 20180317145427.0 cr unu---uuuuu 170323e20111001xx s 000 0 eng 10.1007/s00026-011-0109-2 doi (NATIONALLICENCE)springer-10.1007/s00026-011-0109-2 Complementary Regions of Knot and Link Diagrams [Elektronische Daten] [Colin Adams, Reiko Shinjo, Kokoro Tanaka] An increasing sequence of integers is said to be universal for knots and links if every knot and link has a reduced projection on 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 each universal for knots and links: (3, 5, 7, . . .), (2, n, n+1, n+2, . . .) for each n ≥ 3, (3, n, n+1, n+2, . . .) for each n ≥ 4. Moreover, the finite sequences (2, 4, 5) and (3, 4, n) for each n ≥5 are universal for all knots and links. 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. Springer Basel AG, 2011 knot diagram nationallicence complementary region nationallicence 4-valent graph nationallicence Adams Colin Department of Mathematics and Statistics, Williams College, 18 Hoxsey Street, 01267, Williamstown, MA, USA aut Shinjo Reiko Osaka City University Advanced Mathematical Institute, 3-3-138 Sugimoto, 558-8585, Sumiyoshi-ku, Osaka, Japan aut Tanaka Kokoro Department of Mathematics, Tokyo Gakugei University, 184-8501, Koganei-shi, Tokyo, Japan aut Annals of Combinatorics SP Birkhäuser Verlag Basel 15/4(2011-10-01), 549-563 0218-0006 15:4<549 2011 15 26 https://doi.org/10.1007/s00026-011-0109-2 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00026-011-0109-2 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Adams Colin Department of Mathematics and Statistics, Williams College, 18 Hoxsey Street, 01267, Williamstown, MA, USA aut NATIONALLICENCE 700 1- Shinjo Reiko Osaka City University Advanced Mathematical Institute, 3-3-138 Sugimoto, 558-8585, Sumiyoshi-ku, Osaka, Japan aut NATIONALLICENCE 700 1- Tanaka Kokoro Department of Mathematics, Tokyo Gakugei University, 184-8501, Koganei-shi, Tokyo, Japan aut NATIONALLICENCE 773 0- Annals of Combinatorics SP Birkhäuser Verlag Basel 15/4(2011-10-01), 549-563 0218-0006 15:4<549 2011 15 26 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer