Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-28T14:48:56.308Z Has data issue: false hasContentIssue false
  • English
  • Français

Invariants of tangles

Published online by Cambridge University Press:  24 October 2008

Tim D. Cochran  and
Daniel Ruberman
Tim D. Cochran
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720, U.S.A.
Daniel Ruberman
Affiliation:
Department of Mathematics, Brandeis University, Waltham, MA 02254, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Extract

A tangle is a pair of strings (t0, t1) properly embedded in a 3-ball. Tangles have been used in several approaches to the classification of knots (see [1, 4, 15]). In these investigations, one keeps track of the endpoints of the arcs, so that the sum of two tangles along their boundaries is well defined. In particular, the sum of a given tangle with a trivial tangle, and any invariants of the resulting link, are invariants of the tangle under the restricted relation of isotopy keeping the endpoints fixed.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 105 , Issue 2 , March 1989 , pp. 299 - 306
Copyright
Copyright © Cambridge Philosophical Society 1989

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

REFERENCES

[1]Bonahon, F. and Siebenmann, L.. Geometric Splittings of Knots and 3-Manifolds. London Math. Soc. Lecture Note Ser. (To appear.)Google Scholar
[2]Cochran, T. D.. Concordance invariance of coefficients of Conway's link polynomial. Invent. Math. 82 (1985), 527–541.CrossRefGoogle Scholar
[3]Cochran, T. D.. Geometric invariants of link cobordism. Comment. Math. Helv. 60 (1985), 291–311.CrossRefGoogle Scholar
[4]Conway, J. H.. An enumeration of knots and links, and some of their algebraic properties. In Computational Problems in Abstract Algebra (Pergamon Press, 1970), pp. 329–358.Google Scholar
[5]Cooper, D.. Signatures of surfaces in 3-manifolds and applications to knot and link cobordism. Ph.D. thesis, Warwick University (1982).Google Scholar
[6]Freyd, P., Hoste, J., Lickorish, W. B. R., Millett, K. C., Ocneanu, A. and Yetter, D.. A new polynomial invariant of knots and links. Bull. Amer. Math. Soc. 12 (1985), 239–246.CrossRefGoogle Scholar
[7]Gabai, D.. Foliations and genera of links. Topology1 23 (1984), 381–394.CrossRefGoogle Scholar
[8]Jin, G. T.. Invariants of 2-component links. Ph.D. thesis, Brandeis University (1988).Google Scholar
[9]Kearton, C.. Mutation of knots. (Preprint, 1987.)Google Scholar
[10]Kirk, P.. Link homotopy with one codimension two component. Ph.D. thesis, Brandeis University (1988).Google Scholar
[11]Lickorish, W. B. R. and Millett, K. C.. A polynomial invariant of oriented links. Topology 26 (1987), 107–141.CrossRefGoogle Scholar
[12]Meeks, W. H. and Yau, S. T.. The classical Plateau problem and the topology of three-dimensional manifolds. Topology 21 (1982), 409–440.CrossRefGoogle Scholar
[13]Meyerhoff, R. and Ruberman, D.. Cutting and pasting and the it-invariant. J. Differential Geom. (To appear.)Google Scholar
[14]Milnor, J.. Isotopy of links. In Algebraic Geometry and Topology: Symposium in Honor of S. Lefschetz (Princeton University Press, 1957), 280–306.CrossRefGoogle Scholar
[15]Montesinos, J. M.. Revêtements ramifiés de noeuds, espaces de Seifert et scindements de Heegaard, 1980. (Preprint, 1980.)Google Scholar
[16]Riley, R.. Homomorphisms of knot groups on finite groups. Math. Comp. 25 (1971), 603–619.CrossRefGoogle Scholar
[17]Rolfsen, D.. Knots and Links (Publish or Perish, 1976).Google Scholar
[18]Ruberman, D.. Mutations and volumes of knots in S 3. Invent. Math. 90 (1987), 189–215.CrossRefGoogle Scholar
[19]Sato, N.. Cobordisms of semi-boundary links. Top. Appl. 18 (1984), 225–234.CrossRefGoogle Scholar
[20]Thurston, W. P.. The Geometry and Topology of 3-Manifolds. Lecture notes, Princeton University (1978).Google Scholar