Invariants of piecewise-linear knots

Richard Randell

Banach Center Publications (1998)

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

Abstract

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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

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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

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  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. 

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