Two-Bridge Knots have Residually Finite Groups

On character varieties of two-bridge knot groups

“…From Lemma 5.10 we then conclude v(g n (t 0 )) = 0. Now around P the curve D(k, l) is given by f m+1 (r)/f m (r) = g n+1 (t)/g n (t), which by (19) and (18) of Lemma 5.9 implies…”

“…This implies v(g n (ω)) = 0. 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.…”

Detecting essential surfaces as intersections in the character variety

“…The knot group corresponding to the two-bridge normal form (p, q) has a presentation a, b : aw = wb where a, b are meridians and w = a 1 b 2 · · · a p−2 a p−1 with i = (−1) iq/p and · the floor function (see [17,Proposition 1], [11,Proposition 1]). For this presentation the preferred meridian is given by a and the corresponding preferred longitude is given by ww * a −2e(w) where w * is w written backwards and e(w) = i so that the total exponent sum of the longitude is 0 (see [7, §2]).…”

“…These have knot diagrams as shown in Figure 1 and are obtained as − 1 n and − 1 n surgeries on two components of the Borromean rings as in Figure 2. The first knot in this family, J(4, 4), is the knot 7 4 in the knot tables with two-bridge normal form (15,11).…”

“…The First Knot. The first knot in the family J(2n, 2n) is the knot 7 4 of two-bridge normal form (15,11) with knot group…”