Invariants of piecewise-linear knots

Richard Randell

Banach Center Publications (1998)

  • Volume: 42, Issue: 1, page 307-319
  • ISSN: 0137-6934

Abstract

top
We study numerical and polynomial invariants of piecewise-linear knots, with the goal of better understanding the space of all knots and links. For knots with small numbers of edges we are able to find limits on polynomial or Vassiliev invariants sufficient to determine an exact list of realizable knots. We thus obtain the minimal edge number for all knots with six or fewer crossings. For example, the only knot requiring exactly seven edges is the figure-8 knot.

How to cite

top

Randell, Richard. "Invariants of piecewise-linear knots." Banach Center Publications 42.1 (1998): 307-319. <http://eudml.org/doc/208815>.

@article{Randell1998,
abstract = {We study numerical and polynomial invariants of piecewise-linear knots, with the goal of better understanding the space of all knots and links. For knots with small numbers of edges we are able to find limits on polynomial or Vassiliev invariants sufficient to determine an exact list of realizable knots. We thus obtain the minimal edge number for all knots with six or fewer crossings. For example, the only knot requiring exactly seven edges is the figure-8 knot.},
author = {Randell, Richard},
journal = {Banach Center Publications},
keywords = {PL-knots; PL-invariants; edge number},
language = {eng},
number = {1},
pages = {307-319},
title = {Invariants of piecewise-linear knots},
url = {http://eudml.org/doc/208815},
volume = {42},
year = {1998},
}

TY - JOUR
AU - Randell, Richard
TI - Invariants of piecewise-linear knots
JO - Banach Center Publications
PY - 1998
VL - 42
IS - 1
SP - 307
EP - 319
AB - We study numerical and polynomial invariants of piecewise-linear knots, with the goal of better understanding the space of all knots and links. For knots with small numbers of edges we are able to find limits on polynomial or Vassiliev invariants sufficient to determine an exact list of realizable knots. We thus obtain the minimal edge number for all knots with six or fewer crossings. For example, the only knot requiring exactly seven edges is the figure-8 knot.
LA - eng
KW - PL-knots; PL-invariants; edge number
UR - http://eudml.org/doc/208815
ER -

References

top
  1. [1] D. Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995), 423-472. Zbl0898.57001
  2. [2] J. Birman and X.-S. Lin, Knot polynomials and Vassiliev's invariants, Invent. Math. 111 (1993), 225-270. Zbl0812.57011
  3. [3] G. T. Jin, Polygon indices and superbridge indices, preprint. Zbl0881.57002
  4. [4] G. T. Jin and H. S. Kim, Polygonal knots, J. Korean Math. Soc. 30 (1993), 371-383. 
  5. [5] N. H. Kuiper, A new knot invariant, Math. Ann. 278 (1987), 193-209. Zbl0632.57006
  6. [6] M. Meissen, Edge number results for piecewise-linear knots, this volume. 
  7. [7] K. C. Millett, Knotting of regular Polygons in 3-space, in: Random Knotting and Linking, K. C. Millett and D. W. Sumners (eds.), World Scientific, Singapore, 1994, 31-46. Zbl0838.57008
  8. [8] S. Negami, Ramsey theorems for knots, links and spatial graphs, Trans. Amer. Math. Soc. 324 (1991), 527-541. Zbl0721.57004
  9. [9] R. Randell, A molecular conformation space, in: Proc. 1987 MAT/CHEM/COMP Conference, R. C. Lacher (ed.), Elsevier, 1987. 
  10. [10] S. Negami, Conformation spaces of molecular rings, ibid. 
  11. [11] H. Schubert, Knotten mit zwei Brücken, Math. Z. 65 (1956), 133-170. 
  12. [12] K. Smith, Generalized braid arrangements and related quotient spaces, Univ. of Iowa thesis, 1992. 
  13. [13] T. Stanford, personal communication. 
  14. [14] D. Sumners, New Scientific Applications of Geometry and Topology, Amer. Math. Soc., Providence, 1992. Zbl0759.00015
  15. [15] V. Vassiliev, Cohomology of knot spaces, in: Theory of Singularities and its Applications, V. I. Arnold (ed.), Amer. Math. Soc., Providence, 1990. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.