Acta mathematica scientia,Series B ›› 2012, Vol. 32 ›› Issue (2): 663-671.doi: 10.1016/S0252-9602(12)60047-2

• Articles • Previous Articles     Next Articles

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.

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

CLC Number: 

  • 57M25
Trendmd