Nonabelian representations and signatures of double twist knots

Abstract

Abstract. A conjecture of Riley about the relationship between real parabolic representations and signatures of two-bridge knots is verified for double twist knots.

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

“…One of our interests in the Riley Conjecture is its connection with the question of when the n-fold cyclic branched cover Σ n (K) of a knot K has left-orderable fundamental group. More precisely, as pointed out in [Tr2], Hu's argument in [H] shows that Theorem 1.1 has the following corollary. Corollary 1.2.…”

“…In [R2], Riley states "Some of our computer calculations made in 1972-73 ... suggested that the number of real roots of [λ K (x)] is not less than |σ|/2." Here σ = σ(K) is the signature of K. Following [Tr2], we will refer to this as the Riley Conjecture. The number of real roots of the Riley polynomial of a 2-bridge knot K is at least |σ(K)|/2.…”

“…For double twist knots, the Riley Conjecture was proved by Tran [Tr2]. One of our interests in the Riley Conjecture is its connection with the question of when the n-fold cyclic branched cover Σ n (K) of a knot K has left-orderable fundamental group.…”

Classical pretzel knots and left orderability

“…A sufficient condition for the fundamental group of the n th -cyclic branched cover of S 3 along a prime knot to be left orderable was given in [BGW,Hu1] in terms of SL 2 (R)representations of the knot group. As an application, it was proved in [Go] that for any rational knot K with non-zero signature the fundamental group of the n th -cyclic branched cover of S 3 along K is left orderable for sufficiently large n, see also [Hu1,Tr3]. For a rational knot C(k, l) or C(2n + 1, 2, 2) in the Conway notation, the left orderability of the fundamental groups of the cyclic branched covers of S 3 along the knot was also determined in [DPT,Tr2,Tu].…”