A ZETA-FUNCTION FOR A KNOT USING SL₂(𝔽_p^s) REPRESENTATIONS

An new invariant of a knot, called the zeta-function, is introduced. It is defined by counting the non-diagonalizable representations of the fundamental group into SL₂(𝔽_q), up to conjugacy, and forming a formal power series, similar to a zeta-function from algebraic geometry. The structure of this invariant in the general case is given. The special case of (2, n) torus knots is studied, and basic topological properties are noted.