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

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

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.

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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