Universal Formulae for SU(𝑛) Casson Invariants of Knots

Boden, Hans; Nicas, Andrew

doi:10.1090/S0002-9947-00-02557-5

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.

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