Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-28T00:22:17.490Z Has data issue: false hasContentIssue false
  • English
  • Français

On calculation of the Witten invariants of 3-manifolds

Part of: Low-dimensional topology Topological manifolds

Published online by Cambridge University Press:  09 April 2009

Eugene Rafikov ,
Dušan Repovš  and
Fulvia Spaggiari
Eugene Rafikov
Affiliation:
Department of Differential Geometry, Faculty of Mechanics and Mathematics Moscow State University, Moscow 119899, Russia, e-mail: [email protected]
Dušan Repovš
Affiliation:
Institute for Mathematics Physics and Mechanics University of Ljubljana, P.O. Box 2964, 1001 Ljubljana Slovenia, e-mail: [email protected]
Fulvia Spaggiari
Affiliation:
Department of Mathematics, University of Modena and Reggio EmiliaVia Campi 213/B, 41100 Modena, Italy, e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we present a short definition of the Witten invariants of 3-manifolds. We also give simple proofs of invariance of those obtained for r = 3 and r = 4. Our definition is extracted from the 1993 paper of Lickorish and the Prasolov-Sossinsky book, where it is dispersed over 20 pages. We show by several examples that it is indeed convenient for calculations.

Keywords

Witten invariantKauffman bracketplane diagramDehn surgery3-manifold

MSC classification

Secondary: 57M25: Knots and links in $S^3$ 57N10: Topology of general $3$-manifolds
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 75 , Issue 3 , December 2003 , pp. 385 - 398
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Fenn, R. P. and Rourke, C. P., ‘On Kirby's calculus of links’, Topology 18 (1979), 1–15.CrossRefGoogle Scholar
[2]Kauffman, L. H., ‘State models and the Jones polynomial’, Topology 26 (1987), 395–407.CrossRefGoogle Scholar
[3]Kirby, R., ‘A calculus for framed links in S3’, Invent. Math. (1) 45 (1978), 35–56.CrossRefGoogle Scholar
[4]Kirby, R. and Melvin, P., ‘The 3-manifolds invariants’, Invent. Math. 105 (1991), 473–545.CrossRefGoogle Scholar
[5]Lickorish, W. B. R., ‘Polynomials for links’, Bull. London Math. Soc. 20 (1988), 558–588.CrossRefGoogle Scholar
[6]Lickorish, W. B. R., ‘The skein method for three-manifold invariants’, Pacific J. Math. 149 (1991), 337–347.CrossRefGoogle Scholar
[7]Lickorish, W. B. R., ‘The Skein method for three-manifold invariants’, J. Knot Theory Ramifications 2 (1993), 171–194.CrossRefGoogle Scholar
[8]Prasolov, V. and Sossinsky, A., Knots, links, braids and 3-manifolds, Transl. Math. Monographs 154 (Amer. Math. Soc., Providence, 1997).Google Scholar
[9]Saveliev, N., Lectures on the topology of 3-manifolds (Walter de Gruyter, Berlin, 1999).CrossRefGoogle Scholar