A criterion for satellite 1-genus 1-bridge knots

Abstract

Abstract. Let K be a knot in a closed orientable irreducible 3-manifold M . Suppose M admits a genus 1 Heegaard splitting and we denote by H the splitting torus. We say H is a 1-genus 1-bridge splitting of (M, K) if H intersects K transversely in two points, and divides (M, K) into two pairs of a solid torus and a boundary parallel arc in it. It is known that a 1-genus 1-bridge splitting of a satellite knot admits a satellite diagram disjoint from an essential loop on the splitting torus. If M = S 3 and the slope of the loop is longitudinal in one of the solid tori, then K is obtained by twisting a component of a 2-bridge link along the other component. We give a criterion for determining whether a given 1-genus 1-bridge splitting of a knot admits a satellite diagram of a given slope or not. As an application, we show there exist counter examples for a conjecture of Ait Nouh and Yasuhara.

Knot Floer Homology of (1, 1)-Knots

“…∂[ x 8 ; i, j]). From Figure 10 161 (7), we can verify that the complex CF K ∞ (S 3 , K) is a Z-module generated by…”

“…Recently (1, 1)-knots are extensively studied. See for example, [1], [2], [3], [6], [7], [8], [13], [15], [16], [28] and [29].…”

Dehn Surgeries on 2-Bridge Links Which Yield Reducible 3-Manifolds

“…(1) Theorem 1.3 (1) gives counter examples for Ait-Nouh and Yasuhara's conjecture: if a (p, q)-torus knot (q ≥ p > 0) is obtained by a twisting operation on the trivial knot, then q = np ± 1 for some integer n. See [1]. The existence of the counter examples has already been shown in the previous paper [5]. (2) The essential planar surface corresponding to Theorem 1.2 (3) has 4 boundary circles of slope −1 on a component of the link, and 2 boundary circles of slope −6 on the other component.…”

Thin position for knots, links, and graphs in 3–manifolds

“…The possibility that the companion knot K is a torus knot cannot be dropped; there are satellite knots T in S 3 with b 1 (T ) = 1 [17,20]. For such knots, Morimoto and Sakuma [26] showed that the companion knot K is a torus knot.…”