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Universal Formulae for SU$(n)$ Casson Invariants of Knots


Authors: Hans U. Boden and Andrew Nicas
Journal: Trans. Amer. Math. Soc. 352 (2000), 3149-3187
MSC (1991): Primary 57M25; Secondary 05A19, 14D20, 45G10
DOI: https://doi.org/10.1090/S0002-9947-00-02557-5
Published electronically: March 24, 2000
MathSciNet review: 1695018
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Abstract | References | Similar Articles | Additional Information

Abstract:

An $\operatorname{SU}(n)$ Casson invariant of a knot is an integer which can be thought of as an algebraic-topological count of the number of characters of $\operatorname{SU}(n)$ representations of the knot group which take a longitude into a given conjugacy class. For fibered knots, these invariants can be characterized as Lefschetz numbers which, for generic conjugacy classes, can be computed using a recursive algorithm of Atiyah and Bott, as adapted by Frohman. Using a new idea to solve the Atiyah-Bott recursion (as simplified by Zagier), we derive universal formulae which explicitly compute the invariants for all $n$. Our technique is based on our discovery that the generating functions associated to the relevant Lefschetz numbers (and polynomials) satisfy certain integral equations.


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

  • 1. M. Atiyah and R. Bott, The Yang-Mills equations on a Riemann surface, Phil. Trans. Roy. Soc. Lond. A. 308 (1982), 524-615. MR 85k:14006
  • 2. H. U. Boden, Invariants of fibred knots from moduli, in Geometric Topology, Ed. W. Kazez, AMS/IP Studies in Advanced Mathematics, vol. 2.1, Amer. Math. Soc., 1997, pp. 259-267. MR 98j:57005
  • 3. H. U. Boden, Knot invariants and representation varieties, in preparation, 1999.
  • 4. G. Burde and H Zieschang, Knots, de Gruyter Studies in Mathematics 5, Walter de Gruyter, Berlin, 1985. MR 87b:57004
  • 5. A. Casson, Lecture notes, MSRI Lectures, Berkeley, 1985.
  • 6. U. V. Desale and S. Ramanan, Poincaré polynomials of the variety of stable bundles, Math. Ann. 216 (1975), 233-244. MR 52:402
  • 7. G. P. Egorychev, Integral Representation and the Computation of Combinatorial Sums, AMS Translations of Mathematical Monographs, v. 59, Providence (1984). MR 85a:05008
  • 8. C. Frohman, Unitary representations of knot groups, Topology 32 (1993), 121-144. MR 94g:57003
  • 9. C. Frohman and D. Long, Casson's invariant and surgery on knots, Proc. Edinburgh Math. Soc. 35 (1992), 383-395. MR 94a:57012
  • 10. C. Frohman and A. Nicas, An intersection homology invariant for knots in a rational homology 3-sphere, Topology 33 (1994), 123-158. MR 95c:57010
  • 11. G. Harder and M. S. Narasimhan, On the cohomology groups of moduli spaces of vector bundles over curves, Math. Ann. 212 (1975), 215-248. MR 51:509
  • 12. G. H. Hardy and E. M. Wright, The Theory of Numbers, (1968) fourth edition, Oxford University Press, London.
  • 13. M. Narasimhan and C. Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Ann. Math. 82 (1965), 540-567. MR 32:1725
  • 14. J. Riordan, An Introduction to Combinatorial Analysis, John Wiley, New York, 1958. MR 20:3077
  • 15. D. Zagier, Elementary aspects of the Verlinde formula and of the Harder-Narasimhan-Atiyah-Bott formula, Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), 445-462, Israel Math. Conf. Proc. 9 Bar-Ilan Univ., Ramat Gan, 1996, pp. 445-462. MR 96k:14005

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Additional Information

Hans U. Boden
Affiliation: Department of Mathematics, Ohio State University, Mansfield, Ohio 44906
Email: boden@math.ohio-state.edu

Andrew Nicas
Affiliation: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1
Email: nicas@mcmaster.ca

DOI: https://doi.org/10.1090/S0002-9947-00-02557-5
Keywords: Casson invariants, $\operatorname{SU}(n)$, fibered knots, Alexander polynomial, Conway polynomial, integral equations
Received by editor(s): February 20, 1998
Published electronically: March 24, 2000
Additional Notes: The second-named author was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada
Article copyright: © Copyright 2000 American Mathematical Society

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