We completely determine which Dehn surgeries on 2-bridge links yield reducible 3-manifolds. Further, we consider which surgery on one component of a 2-bridge link yields a torus knot, a cable knot and a satellite knot in this paper.
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.
ABSTRACT. In this paper we utilize BP* ( ) , a generalized cohomology theory associated with the Brown-Peterson spectrum to prove a nonimmersion theorem for products of real projective spaces.
STATEMENT OF RESULTSLet v(2a(2b + 1)) = a and let o:(n) denote the number of l's in the binary expansion of n . Let pn denote the n-dimensional real projective space. Then our main theorem is 4(m, +,,·+md-2d-k-1 R .To our knowledge, the only published works on non immersions of products of real projective spaces are those of Suzuki [18] and Kobayashi [12]. Their methods seem to be effective for pm X pn with m = 2' -1 and n = 2 s -1 .When s > r > 7, their nonimmersions are in dimensions m + n + 2 s -1 (see [12, Corollary 4]
Which slopes can or cannot appear as Seifert fibered slopes for hyperbolic knots in the 3-sphere S 3 ? It is conjectured that if r -surgery on a hyperbolic knot in S 3 yields a Seifert fiber space, then r is an integer. We show that for each integer n ∈ Z, there exists a tunnel number one, hyperbolic knot K n in S 3 such that n-surgery on K n produces a small Seifert fiber space.
There are exactly four mutually non-isotopic unknotting tunnels τi, i = 1,2,3,4 for the pretzel knot P(-2,3,7). Moreover, there are at most 3 non-stabilized genus 3 Heegaard splittings.
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