Parabolic Representations of Knot Groups, I

Commensurability classes of 2–bridge knot complements

“…Hence we exclude this case from further comment. Riley proves the following result which will be useful for us [21,Theorem 3].…”

“…In [21], Riley also describes the image of a longitude`for x 1 for a p-rep in standard form; namely a matrix…”

“…then Riley shows that y satisfies a certain polynomial ƒ K .y/ with leading coefficient and constant term equal to 1 [21,Theorem 2] . We shall say that the above p-rep is in standard form.…”

Nonabelian representations and signatures of double twist knots

“…The zero set in C of the polynomial Φ K (2, y) of a two-bridge knot K describes the set of all parabolic representations of the knot group of K into SL 2 (C), see [Ri2]. Riley made the following conjecture about the relationship between the number of real roots of Φ K (2, y) and the signature σ(K) of K.…”

Riley's conjecture on SL(2,R) representations of 2-bridge knots

“…In particular, he considered non-abelian representations in which the meridians go to parabolic elements, calling these parabolic representations. He showed that, for a given 2-bridge knot K, such representations correspond to the roots of a certain polynomial λ K (x) ∈ Z[x], the Riley polynomial; see [R1,Theorem 2]. Thus the real roots of λ K (x) give parabolic SL(2, R) representations.…”