Universal Formulae for SU ( n ) (n) Casson Invariants of Knots

An SU⁡(n)\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 SU⁡(n)\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 nn. Our technique is based on our discovery that the generating functions associated to the relevant Lefschetz numbers (and polynomials) satisfy certain integral equations.