Non-acyclic SL₂-representations of Twist Knots, -3-Dehn Surgeries, and L-functions

We study irreducible SL2-representations of twist knots. We first determine all nonacyclic SL2(C)-representations, which turn out to lie on a line denoted as x = y in R2. Our main tools are character variety, Reidemeister torsion, and Chebyshev polynomials. We also verify a certain common tangent property, which yields a result on L-functions, that is, the orders of the knot modules associated to the universal deformations. Secondly, we prove that a representation is on the line x = y if and only if it factors through the -3-Dehn surgery, and is non-acyclic if and only if the image of a certain element is of order 3. Finally, we study absolutely irreducible nonacyclic representations over a finite field with characteristic p < 2, to concretely determine all non-trivial L-functions L? of the universal deformations over complete discrete valuation rings. We show among other things that L = kn(x)2 holds for a certain series kn(x) of polynomials.