Two-bridge knots have residually finite groups

On character varieties of two-bridge knot groups

“…Proof. From (19) we deduce h 2n (ω) 2 − 1 − 3n = ng n (ω) −2 A with A = −2(ω + 4) + 3(4 − g n (ω) 2 ). By Lemma 5.10 we have v(g n (ω)) = 0, so v(h 2n (ω) 2 − 1 − 3n) = v(n) + v(A).…”

“…This presentation follows from the canonical Schubert normal form [30] of the two-bridge diagram of K(p, q) (see [25,Prop. 1], [20, (2.1)], [19,Prop. 1]).…”

“…From Lemma 5.10 we find that if ω is a root of G n , and v is some extension of the valuation associated to a prime dividing 2n, then the valuation at v of the element in (19) equals v(2n) + v(ω − 2). The proof of the following proposition shows that for odd k, in order to show that D(k, 2n) is smooth, it suffices to note that this valuation is at least 1.…”

“…The only remaining case is the case that k is even and kl is positive. We deal with this case by investigating the possible values of the valuation of the expression in (19) at some valuation extending v p for some prime p dividing 2n. Lemma 5.12.…”

“…By Lemma 5.4(2) the root ω is an algebraic integer, so we have v(2 − ω) ≥ 0 and v(g n ≥ 0. From (19)…”