Universal Formulae for SU$(n)$ Casson Invariants of Knots
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- by Hans U. Boden and Andrew Nicas PDF
- Trans. Amer. Math. Soc. 352 (2000), 3149-3187 Request permission
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
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Additional Information
- Hans U. Boden
- Affiliation: Department of Mathematics, Ohio State University, Mansfield, Ohio 44906
- MR Author ID: 312802
- ORCID: 0000-0001-5516-8327
- Email: boden@math.ohio-state.edu
- Andrew Nicas
- Affiliation: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1
- MR Author ID: 131000
- Email: nicas@mcmaster.ca
- 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
- © Copyright 2000 American Mathematical Society
- 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
- MathSciNet review: 1695018