Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Splicing and the $ SL_2({\mathbb{C}})$ Casson invariant


Authors: Hans U. Boden and Cynthia L. Curtis
Journal: Proc. Amer. Math. Soc. 136 (2008), 2615-2623
MSC (2000): Primary 57M27; Secondary 57M25, 57M05
DOI: https://doi.org/10.1090/S0002-9939-08-09233-2
Published electronically: March 14, 2008
MathSciNet review: 2390534
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We establish a formula for the $ SL_2({\mathbb{C}})$ Casson invariant of spliced sums of homology spheres along knots. Along the way, we show that the $ SL_2({\mathbb{C}})$ Casson invariant vanishes for spliced sums along knots in $ S^3$.


References [Enhancements On Off] (What's this?)

  • 1. H. U. Boden and C. L. Curtis, The $ SL_2({\mathbb{C}})$ Casson invariant for Seifert fibered homology spheres and surgeries on twist knots, J. Knot Theory Ramifications 15 (2006), 813-837. MR 2264157 (2007g:57017)
  • 2. S. Boyer and A. Nicas, Varieties of group representations and Casson's invariant for rational homology $ 3$-spheres, Trans. Amer. Math. Soc. 322 (1990), 507-522. MR 972701 (92a:57020)
  • 3. S. Boyer and X. Zhang, On Culler-Shalen seminorms and Dehn filling, Annals of Math. 148 (1998), 737-801. MR 1670053 (2000d:57028)
  • 4. S. Boyer and X. Zhang, A proof of the finite filling conjecture, J. Diff. Geom. 59 (2001), 87-176. MR 1909249 (2003k:57007)
  • 5. S. Boyer and X. Zhang, Every nontrivial knot in $ S^3$ has nontrivial $ A$-polynomial, Proc. Amer. Math. Soc. 133 (2005), 2813-2815. MR 2146231 (2006g:57018)
  • 6. D. Cooper, M. Culler, H. Gillet, D. D. Long, and P. B. Shalen, Plane curves associated to character varieties of $ 3$-manifolds, Invent. Math. 118 (1994), 47-84. MR 1288467 (95g:57029)
  • 7. C. L. Curtis, An intersection theory count of the $ \SLC$-representations of the fundamental group of a $ 3$-manifold, Topology 40 (2001), 773-787. MR 1851563 (2002k:57022)
  • 8. C. L. Curtis, Erratum to ``An intersection theory count of the $ SL_2({\mathbb{C}})$ representations of the fundamental group of a $ 3$-manifold,'' Topology 42 (2003), 929. MR 1958534 (2004b:57018)
  • 9. N. M. Dunfield and S. Garoufalidis, Nontrviality of the $ A$-polynomial for knots in $ S^3$, Algebr. Geom. Topol. 4 (2004), 1145-1153. MR 2113900 (2005i:57004)
  • 10. S. Fukuhara and N. Maruyama, A sum formula for Casson's $ \lambda$-invariant, Tokyo J. Math. 11 (1988), 281-287. MR 976563 (89k:57021)
  • 11. J. Hempel, $ 3$-Manifolds, Annals of Mathematics Studies 86, Princeton University Press, Princeton, NJ (1976). MR 0415619 (54:3702)
  • 12. P. Kirk, E. Klassen and D. Ruberman, Splitting the spectral flow and the Alexander matrix, Comm. Math. Helv. 69 (1994), 375-416. MR 1289334 (95h:57037)
  • 13. P. B. Kronheimer and T. S. Mrowka, Witten's conjecture and property P, Geom. Topol. 8 (2004), 295-310. MR 2023280 (2004m:57023)
  • 14. P. B. Kronheimer and T. S. Mrowka, Dehn surgery, the fundamental group and SU$ (2)$, Math. Res. Lett. 11 (2004), 741-754. MR 2106239 (2005k:57018)
  • 15. N. Saveliev, Invariants for homology $ 3$-spheres, Encyclopaedia of Math. Sci. 140, Springer-Verlag, Berlin, Heidelberg (2002). MR 1941324 (2004c:57026)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 57M27, 57M25, 57M05

Retrieve articles in all journals with MSC (2000): 57M27, 57M25, 57M05


Additional Information

Hans U. Boden
Affiliation: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1 Canada
Email: boden@mcmaster.ca

Cynthia L. Curtis
Affiliation: Department of Mathematics and Statistics, The College of New Jersey, Ewing, New Jersey 08628
Email: ccurtis@tcnj.edu

DOI: https://doi.org/10.1090/S0002-9939-08-09233-2
Keywords: Casson invariant, character variety, spliced sum.
Received by editor(s): March 28, 2007
Published electronically: March 14, 2008
Additional Notes: The first named author was supported by a grant from the Natural Sciences and Engineering Research Council of Canada.
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society