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.
@article{bwmeta1.element.bwnjournal-article-bcpv42i1p307bwm,
author = {Randell, Richard},
title = {Invariants of piecewise-linear knots},
journal = {Banach Center Publications},
volume = {43},
year = {1998},
pages = {307-319},
zbl = {0901.57014},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv42i1p307bwm}
}
Randell, Richard. Invariants of piecewise-linear knots. Banach Center Publications, Tome 43 (1998) pp. 307-319. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv42i1p307bwm/
[000] [1] D. Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995), 423-472. | Zbl 0898.57001
[001] [2] J. Birman and X.-S. Lin, Knot polynomials and Vassiliev's invariants, Invent. Math. 111 (1993), 225-270. | Zbl 0812.57011
[002] [3] G. T. Jin, Polygon indices and superbridge indices, preprint. | Zbl 0881.57002
[003] [4] G. T. Jin and H. S. Kim, Polygonal knots, J. Korean Math. Soc. 30 (1993), 371-383.
[004] [5] N. H. Kuiper, A new knot invariant, Math. Ann. 278 (1987), 193-209. | Zbl 0632.57006
[005] [6] M. Meissen, Edge number results for piecewise-linear knots, this volume.
[006] [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. | Zbl 0838.57008
[007] [8] S. Negami, Ramsey theorems for knots, links and spatial graphs, Trans. Amer. Math. Soc. 324 (1991), 527-541. | Zbl 0721.57004
[008] [9] R. Randell, A molecular conformation space, in: Proc. 1987 MAT/CHEM/COMP Conference, R. C. Lacher (ed.), Elsevier, 1987.
[009] [10] S. Negami, Conformation spaces of molecular rings, ibid.
[010] [11] H. Schubert, Knotten mit zwei Brücken, Math. Z. 65 (1956), 133-170.
[011] [12] K. Smith, Generalized braid arrangements and related quotient spaces, Univ. of Iowa thesis, 1992.
[012] [13] T. Stanford, personal communication.
[013] [14] D. Sumners, New Scientific Applications of Geometry and Topology, Amer. Math. Soc., Providence, 1992. | Zbl 0759.00015
[014] [15] V. Vassiliev, Cohomology of knot spaces, in: Theory of Singularities and its Applications, V. I. Arnold (ed.), Amer. Math. Soc., Providence, 1990.