Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Complementary Regions of Knot and Link Diagrams

  • 111 Accesses

  • 11 Citations

Abstract

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.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Eberhard V.: Zur morphologie der polyeder. Teubner, Leipzig (1891)

  2. 2.

    Eliahou S., Harary F., Kauffman L.: Lune-free knot graphs. J. Knot Theory Ramifications 17(1), 55–74 (2008)

  3. 3.

    Enns T.C.: 4-valent graphs. J. Graph Theory 6(3), 255–281 (1982)

  4. 4.

    Grünbaum B.: Some analogues of Eberhard’s theorem on convex polytopes. Israel J. Math. 6(4), 398–411 (1968)

  5. 5.

    Hoste J., Thistlethwaite M., Weeks J.: The first 1,701,936 knots. Math. Intelligencer 20(4), 33–48 (1998)

  6. 6.

    Hotz G.: Arkadenfadendarstellung von Knoten und eine neue Darstellung der Knotengruppe. Abh. Math. Sem. Univ. Hamburg 24, 132–148 (1960)

  7. 7.

    Jeong D.: Realizations with a cut-through Eulerian circuit. Discrete Math. 137(1-3), 265–275 (1995)

  8. 8.

    Kauffman L.: State models and the Jones polynomial. Topology 26(3), 395–407 (1987)

  9. 9.

    Lee J.H., Jin G.T.: Link diagrams realizing prescribed subdiagram partitions. Kobe J. Math. 18(2), 199–202 (2001)

  10. 10.

    Murasugi K.: Jones polynomials and classical conjectures in knot theory. Topology 26(2), 187–194 (1987)

  11. 11.

    Ozawa, M.: Edge number of knots and links. Preprint at arXiv:0705.4348. (2007)

  12. 12.

    Shinjo, R.: Complementary Regions of Projections of Spatial Graphs. In preparation.

  13. 13.

    Thistlethwaite M.: A spanning tree expansion of the Jones polynomial. Topology 26(3), 297–309 (1987)

Download references

Author information

Correspondence to Colin Adams.

Additional information

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).

Rights and permissions

Reprints and Permissions

About this article

Cite this article

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

Download citation

Mathematics Subject Classification

  • 57M25
  • 5C10
  • 57M15

Keywords

  • knot diagram
  • complementary region
  • 4-valent graph