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Complementary Regions of Knot and Link Diagrams
Colin Adams1, Reiko Shinjo2, and Kokoro Tanaka3
1Department ofMathematics and Statistics,Williams College, 18 Hoxsey Street,Williamstown, MA 01267, USA
2Osaka City University Advanced Mathematical Institute, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan
Department of Mathematics, Tokyo Gakugei University, Koganei-shi, Tokyo 184-8501, Japan
Annals of Combinatorics 15 (4) pp.549-563 October, 2011
AMS Subject Classification: 57M25, 5C10; 57M15
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.
Keywords: knot diagram, complementary region, 4-valent graph


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