Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T05:52:17.248Z Has data issue: false hasContentIssue false
  • English
  • Français

Upper bound on lattice stick number of knots

Published online by Cambridge University Press:  25 April 2013

KYUNGPYO HONG ,
SUNGJONG NO  and
SEUNGSANG OH
KYUNGPYO HONG
Affiliation:
Department of Mathematics, Korea University, Seoul 136-701, Korea. e-mails: cguyhbjm@korea.ac.kr, blueface@korea.ac.kr, seungsang@korea.ac.kr
SUNGJONG NO
Affiliation:
Department of Mathematics, Korea University, Seoul 136-701, Korea. e-mails: cguyhbjm@korea.ac.kr, blueface@korea.ac.kr, seungsang@korea.ac.kr
SEUNGSANG OH
Affiliation:
Department of Mathematics, Korea University, Seoul 136-701, Korea. e-mails: cguyhbjm@korea.ac.kr, blueface@korea.ac.kr, seungsang@korea.ac.kr
Get access Rights & Permissions [Opens in a new window]

Abstract

The lattice stick number sL(K) of a knot K is defined to be the minimal number of straight line segments required to construct a stick presentation of K in the cubic lattice. In this paper, we find an upper bound on the lattice stick number of a nontrivial knot K, except the trefoil knot, in terms of the minimal crossing number c(K) which is sL(K) ≤ 3c(K) + 2. Moreover if K is a non-alternating prime knot, then sL(K) ≤ 3c(K) − 4.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 155 , Issue 1 , July 2013 , pp. 173 - 179
Copyright
Copyright © Cambridge Philosophical Society 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Adams, C.The Knot Book. W.H. Freedman & Co., New York, 1994.Google Scholar
[2]Adams, C., Chu, M., Crawford, T., Jensen, S., Siegel, K. and Zhang, L.Stick index of knots and links in the cubic lattice. J. Knot Theory Ramifications 21 (2012) no. 5, 16 pp.CrossRefGoogle Scholar
[3]Bae, Y. and Park, C. Y.An upper bound of arc index of links. Math. Proc. Camb. Phil. Soc. 129 (2000), 491–500.CrossRefGoogle Scholar
[4]Calvo, J. A.Characterizing polygons in ℝ3, Physical knots: knotting, linking, and folding geometric objects in ℝ3 (Las Vegas, NY, 2001). Contemp. Math. 304 (2002), 37–53.CrossRefGoogle Scholar
[5]Cromwell, P.Embedding knots and links in an open book I: Basic Properties. Topology Appl. 64 (1995), 37–58.CrossRefGoogle Scholar
[6]Diao, Y.Minimal knotted polygons on the cubic lattice. J. Knot Theory Ramifications 2 (1993), 413–425.CrossRefGoogle Scholar
[7]Diao, Y.The number of smallest knots on the cubic lattice. J. Stat. Phys. 74 (1994), 1247–1254.CrossRefGoogle Scholar
[8]Elifai, E. A.On stick numbers of knots and links. Chaos, Solitons and Fractals 27 (2006), 233–236.CrossRefGoogle Scholar
[9]Furstenberg, E., Li, J. and Schneider, J.Stick knots. Chaos, Solitons and Fractals 9 (1998), 561–568.CrossRefGoogle Scholar
[10]Huh, Y. and Oh, S.Lattice stick numbers of small knots. J. Knot Theory Ramifications 14 (2005), 859–867.CrossRefGoogle Scholar
[11]Huh, Y. and Oh, S.Knots with small lattice stick numbers. J. Phys. A: Math. Theor. 43 (2010), 265002(8pp).CrossRefGoogle Scholar
[12]Huh, Y. and Oh, S.An upper bound on stick number of knots. J. Knot Theory Ramifications 20 (2011), 741–747.CrossRefGoogle Scholar
[13]Janse van Rensburg, E. J. and Promislow, S. D.The curvature of lattice knots. J. Knot Theory Ramifications 8 (1999), 463–490.CrossRefGoogle Scholar
[14]Jin, G. T. and Park, W. K.Prime knots with arc index up to 11 and an upper bound of arc index for non-alternating knots. J. Knot Theory Ramifications 19 (2010), 1655–1672.CrossRefGoogle Scholar
[15]Negami, S.Ramsey theorems for knots, links and spatial graphs. Trans. Amer. Math. Soc. 324 (1991), 527–541.CrossRefGoogle Scholar