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.
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Dedicated to Akio Kawauchi on the occasion of his 60th birthday
The author is partially supported by NSF Grant DMS-0306211.
The author is grateful to Toshiki Endo, Kouki Taniyama and Akira Yasuhara for helpful comments.
The author is grateful to Toshiki Endo, Kouki Taniyama and Akira Yasuhara for helpful comments, and is partially supported by Grant-in-Aid for Young Scientists (B) (No.19740041).
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Adams, C., Shinjo, R. & Tanaka, K. Complementary Regions of Knot and Link Diagrams. Ann. Comb. 15, 549–563 (2011). https://doi.org/10.1007/s00026-011-0109-2
Mathematics Subject Classification
- knot diagram
- complementary region
- 4-valent graph