数学物理学报(英文版) ›› 2012, Vol. 32 ›› Issue (2): 663-671.doi: 10.1016/S0252-9602(12)60047-2

• 论文 • 上一篇    下一篇

SOME APPLICATIONS OF PLANAR GRAPH IN KNOT THEORY

程志云|高红铸   

  1. School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China;Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China
  • 收稿日期:2009-12-22 修回日期:2010-09-02 出版日期:2012-03-20 发布日期:2012-03-20
  • 基金资助:

    Supported by NSF 10671018 and NSF 11171025.

SOME APPLICATIONS OF PLANAR GRAPH IN KNOT THEORY

 CHENG Zhi-Yun, GAO Hong-Zhu   

  1. School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China;Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China
  • Received:2009-12-22 Revised:2010-09-02 Online:2012-03-20 Published:2012-03-20
  • Supported by:

    Supported by NSF 10671018 and NSF 11171025.

摘要:

The relationship between a link diagram and its corresponding planar graph is briefly reviewed. A necessary and sufficient condition is given to detect when a planar graph corresponds to a knot. The relationship between planar graph and almost planar Seifert surface is discussed. Using planar graph, we construct an alternating amphicheiral prime knot with crossing number n for any even number n  4. This gives an affirmative answer to problem 1.66(B) on Kirby’s problem list.

关键词: Planar graph, almost planar Seifert surface, amphicheiral knot, alternating knot

Abstract:

The relationship between a link diagram and its corresponding planar graph is briefly reviewed. A necessary and sufficient condition is given to detect when a planar graph corresponds to a knot. The relationship between planar graph and almost planar Seifert surface is discussed. Using planar graph, we construct an alternating amphicheiral prime knot with crossing number n for any even number n  4. This gives an affirmative answer to problem 1.66(B) on Kirby’s problem list.

Key words: Planar graph, almost planar Seifert surface, amphicheiral knot, alternating knot

中图分类号: 

  • 57M25