Download PDF - Invariants of piecewise-linear knotsAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Invariants of piecewise-linear knots

Authors 1

Affiliations

  1. Department of Mathematics, University of Iowa, Iowa City, Iowa 52242, U.S.A.

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.

Bibliography

  1. D. Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995), 423-472.
  2. J. Birman and X.-S. Lin, Knot polynomials and Vassiliev's invariants, Invent. Math. 111 (1993), 225-270.
  3. G. T. Jin, Polygon indices and superbridge indices, preprint.
  4. G. T. Jin and H. S. Kim, Polygonal knots, J. Korean Math. Soc. 30 (1993), 371-383.
  5. N. H. Kuiper, A new knot invariant, Math. Ann. 278 (1987), 193-209.
  6. M. Meissen, Edge number results for piecewise-linear knots, this volume.
  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.
  8. S. Negami, Ramsey theorems for knots, links and spatial graphs, Trans. Amer. Math. Soc. 324 (1991), 527-541.
  9. R. Randell, A molecular conformation space, in: Proc. 1987 MAT/CHEM/COMP Conference, R. C. Lacher (ed.), Elsevier, 1987.
  10. S. Negami, Conformation spaces of molecular rings, ibid.
  11. H. Schubert, Knotten mit zwei Brücken, Math. Z. 65 (1956), 133-170.
  12. K. Smith, Generalized braid arrangements and related quotient spaces, Univ. of Iowa thesis, 1992.
  13. T. Stanford, personal communication.
  14. D. Sumners, New Scientific Applications of Geometry and Topology, Amer. Math. Soc., Providence, 1992.
  15. V. Vassiliev, Cohomology of knot spaces, in: Theory of Singularities and its Applications, V. I. Arnold (ed.), Amer. Math. Soc., Providence, 1990.
Banach Center Publications
1998/42/1
Export to PBN, Opens in new tab
Pages:
307-319
Main language of publication
English
Published
1998
Exact and natural sciences