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Exact number of chord diagrams and an estimation of the number of spine diagrams of ordern

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Chinese Science Bulletin

Abstract

The main results are as follows:

  1. (I)

    For the number of chord diagram of ordern, an exact formula is given.

  2. (II)

    For the number of spine diagrams of ordern, the upper and lower bounds are obtained. These bounds show that the estimation is asymptotically the best.

As a byproduct, an upper bound is obtained, for the dimension of Vassiliev knot invariants of ordern, that is, 1/2(n − 1)! for anyn≥3, and 1/2 (n−1)! −1/2 (n−2)! for biggern. Our upper bound is based on the work of Chmutov and Duzhin and is an improvement of their bound (n − 1)!. Forn = 3, and 4, 1/2(n − 1)! is already the best.

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References

  1. Arnold, V. I., Vassiliev’s theory of discriminants and knots,Plenary Lecture at the First European Congress of Mathematics. Paris, July, 1992.

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Authors and Affiliations

  1. Institute of Systems Science, Chinese Academy of Sciences, 100080, Beijing, China

    Banghe Li & Hongwei Sun

Authors
  1. Banghe Li
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  2. Hongwei Sun
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Additional information

The first version of this paper was the preprint titled “The exact number of chord diagrams and an upper bound of Vassiliev knot invariants of ordern”, which was typed in January 1996. and included in the “Bibliography of Vassiliev Invariants” with last updated: Mon Jun 24 09:41:28 IDT 1996. edited by Bar-Natan.

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Li, B., Sun, H. Exact number of chord diagrams and an estimation of the number of spine diagrams of ordern . Chin.Sci.Bull. 42, 705–718 (1997). https://doi.org/10.1007/BF03186960

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  • DOI: https://doi.org/10.1007/BF03186960

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