No Access

AVERAGE CROSSING NUMBER, TOTAL CURVATURE AND ROPELENGTH OF THICK KNOTS

C. ERNST

Department of Mathematics and Computer Science, Western Kentucky University, Bowling Green, KY 42101, USA

Search for more papers by this author

and

A. POR

Department of Mathematics and Computer Science, Western Kentucky University, Bowling Green, KY 42101, USA

Search for more papers by this author

https://doi.org/10.1142/S0218216511009601Cited by:3 (Source: Crossref)

Abstract

Let K be a smooth knot of unit thickness embedded in the space with length L(K) and total curvature κ(K). Then where acn(K) is the average crossing number of the embedded knot K and c > 0 is a constant independent of the knot K. This relationship had been conjectured in [G. Buck and J. Simon, Total curvature and packing of knots, Topology Appl.154 (2007) 192–204] where it is shown that the square root power on the curvature is the lowest possible. In the last section we give several examples to illustrate some relationships between the three quantities average crossing number, total curvature and ropelength.

Keywords:

AMSC: 57M25, 57M27

References

G. Buck and J. Simon, Topology Appl. 154, 192 (2007), DOI: 10.1016/j.topol.2006.04.002. Crossref, Web of Science, Google Scholar
G. Burde and H. Zieschang , Knots , 2nd edn. ( Walter de Gruyter , 2002 ) . Crossref, Google Scholar
J. Cantarella, R. B. Kusner and J. M. Sullivan, vent. Math. 15(2), 257 (2002). Google Scholar
Y. Diao and C. Ernst, Math. Proc. Cambridge Philos. Soc. 143, 41 (2007). Crossref, Web of Science, Google Scholar
Y. Diao, C. Ernst and E. J. Janse Van Rensburg, Math. Proc. Cambridge Philos. Soc. 126, 293 (1999). Crossref, Web of Science, Google Scholar
Y. Diao, C. Ernst and E. J. Janse Van Rensburg, J. Knot Theory Ramifications 11, 199 (2002). Link, Web of Science, Google Scholar
Y. Diao, C. Ernst and M. Thistlethwaite, J. Knot Theory and Ramifications 12(5), 709 (2003), DOI: 10.1142/S0218216503002615. Link, Web of Science, Google Scholar
C. Ernst and D. W. Sumners, Math. Proc. Cambridge Philos. Soc. 102, 303 (1987), DOI: 10.1017/S0305004100067323. Crossref, Web of Science, Google Scholar
A. Gray , Tubes , 2nd edn. , Progress in Mathematics 221 ( Birkhäuser Verlag , Basel , 2004 ) . Crossref, Google Scholar
K. V. Kleninet al., J. Biomol. Struct. Dyn. 5, 1173 (1988). Crossref, Web of Science, Google Scholar
R. Litherlandet al., Topology Appl. 91(3), 233 (1999), DOI: 10.1016/S0166-8641(97)00210-1. Crossref, Web of Science, Google Scholar
C. Sundberg and M. B. Thistlethwaite, Pacific J. Math. 182(2), 329 (1998), DOI: 10.2140/pjm.1998.182.329. Crossref, Web of Science, Google Scholar