Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-11T06:42:38.758Z Has data issue: false hasContentIssue false
  • English
  • Français

Lonely knots and tangles: Identifying knots with no companions

Part of: Low-dimensional topology

Published online by Cambridge University Press:  26 February 2010

P. R. Cromwell
P. R. Cromwell
Affiliation:
School of Mathematics, University College of North Wales, Dean Street, Bangor, Gwynedd. LL57 1UT
Get access

Extract

There are two principal ways of decomposing knots and links into simpler ones: (1) a sphere intersecting the knot in two points gives a connected sum decomposition; (2) an incompressible torus in the knot complement gives a satellite decomposition. If a knot K is such that in every connected sum decomposition one of the factors is an unknotted arc spanning the sphere then K is called a prime knot. In [L] Raymond Lickorish explored the possibility of using 2-string tangles to construct and detect prime knots. He defined prime tangles and showed that the sum of prime tangles is always a prime knot or link. Later, Quach Cam Van studied partial sums of tangles and gave necessary and sufficient conditions for the resulting tangle to be prime. In this paper, similar results are established which relate to the satellite decomposition rather than to the connected sum.

MSC classification

Secondary: 57M25: Knots and links in $S^3$
Type
Research Article
Information
Mathematika , Volume 38 , Issue 2 , December 1991 , pp. 334 - 347
Copyright
Copyright © University College London 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Co.Conway, J. H.. An enumeration of knots and links and some of their related properties. Computational Problems in Abstract Algebra, (1969), 329–358.CrossRefGoogle Scholar
Crl.Cromwell, P. R.. Homogeneous links. J. London Math. Soc. (2), 39 (1989), 535–552.CrossRefGoogle Scholar
Cr2.Cromwell, P. R.. Some infinite families of satellite knots with given Alexander polynomial. Mathematika, 38 (1991), 156–169.CrossRefGoogle Scholar
K.Kauffman, L. H.. State models and the Jones polynomial. Topology, 26 (1987), 395–407.CrossRefGoogle Scholar
L.Lickorish, W. B. R.. Prime knots and tangles. Trans. Amer. Math. Soc, 267 (1981), 321–332.CrossRefGoogle Scholar
Me.Menasco, W.. Closed incompressible surfaces in alternating knot and link complements. Topology, 23 (1984), 37–44.CrossRefGoogle Scholar
Mu.Murasugi, K.. Jones polynomials and classical conjectures in knot theory. Topology, 26 (1987), 187–194.CrossRefGoogle Scholar
Q.Quach Thi Cam Van. On a theorem on partially summing tangles by Lickorish. Math. Proc. Cambridge Phil. Soc, 93 (1983), 63–66.CrossRefGoogle Scholar
Tl.Thistlethwaite, M. B.. A spanning tree expansion of the Jones polynomial. Topology, 26 (1987), 297–309.CrossRefGoogle Scholar
T2.Thistlethwaite, M. B.. On the Kauffman polynomial of an adequate link. Invent. Math., 93 (1988), 285–296.CrossRefGoogle Scholar
T3.Thistlethwaite, M. B.. An upper bound for the breadth of the Jones polynomial. Math. Proc. Cambridge Phil. Soc, 103 (1988), 451–456.CrossRefGoogle Scholar