Universal Formulae for SU 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

Published electronically:
March 24, 2000

MathSciNet review:
1695018

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Abstract | References | Similar Articles | Additional Information

An Casson invariant of a knot is an integer which can be thought of as an algebraic-topological count of the number of characters of 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 . Our technique is based on our discovery that the generating functions associated to the relevant Lefschetz numbers (and polynomials) satisfy certain integral equations.

**1.**M. F. Atiyah and R. Bott,*The Yang-Mills equations over Riemann surfaces*, Philos. Trans. Roy. Soc. London Ser. A**308**(1983), no. 1505, 523–615. MR**702806**, 10.1098/rsta.1983.0017**2.**Hans U. Boden,*Invariants of fibred knots from moduli*, Geometric topology (Athens, GA, 1993) AMS/IP Stud. Adv. Math., vol. 2, Amer. Math. Soc., Providence, RI, 1997, pp. 259–267. MR**1470731****3.**H. U. Boden,*Knot invariants and representation varieties,*in preparation, 1999.**4.**Gerhard Burde and Heiner Zieschang,*Knots*, de Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter & Co., Berlin, 1985. MR**808776****5.**A. Casson, Lecture notes, MSRI Lectures, Berkeley, 1985.**6.**Usha V. Desale and S. Ramanan,*Poincaré polynomials of the variety of stable bundles*, Math. Ann.**216**(1975), no. 3, 233–244. MR**0379497****7.**G. P. Egorychev,*Integral representation and the computation of combinatorial sums*, Translations of Mathematical Monographs, vol. 59, American Mathematical Society, Providence, RI, 1984. Translated from the Russian by H. H. McFadden; Translation edited by Lev J. Leifman. MR**736151****8.**Charles Frohman,*Unitary representations of knot groups*, Topology**32**(1993), no. 1, 121–144. MR**1204411**, 10.1016/0040-9383(93)90042-T**9.**C. D. Frohman and D. D. Long,*Casson’s invariant and surgery on knots*, Proc. Edinburgh Math. Soc. (2)**35**(1992), no. 3, 383–395. MR**1187001**, 10.1017/S0013091500005678**10.**Charles Frohman and Andrew Nicas,*An intersection homology invariant for knots in a rational homology 3-sphere*, Topology**33**(1994), no. 1, 123–158. MR**1259519**, 10.1016/0040-9383(94)90039-6**11.**G. Harder and M. S. Narasimhan,*On the cohomology groups of moduli spaces of vector bundles on curves*, Math. Ann.**212**(1974/75), 215–248. MR**0364254****12.**G. H. Hardy and E. M. Wright, The Theory of Numbers, (1968) fourth edition, Oxford University Press, London.**13.**M. S. Narasimhan and C. S. Seshadri,*Stable and unitary vector bundles on a compact Riemann surface*, Ann. of Math. (2)**82**(1965), 540–567. MR**0184252****14.**John Riordan,*An introduction to combinatorial analysis*, Wiley Publications in Mathematical Statistics, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. MR**0096594****15.**Don 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) Israel Math. Conf. Proc., vol. 9, Bar-Ilan Univ., Ramat Gan, 1996, pp. 445–462. MR**1360519**

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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