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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
Published electronically: March 24, 2000
MathSciNet review: 1695018
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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.


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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: http://dx.doi.org/10.1090/S0002-9947-00-02557-5
PII: S 0002-9947(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