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On the geometric and algebraic rank of graph manifolds
, 2003
"... Abstract. For any n ∈ N we construct graph manifolds of genus 4n that have 3ngenerated fundamental group. 1. introduction A Heegaard surface of an orientable closed 3manifold M is an embedded orientable surface S such that M − S consists of 2 handlebodies V1 and V2. This decomposition of M is call ..."
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Abstract. For any n ∈ N we construct graph manifolds of genus 4n that have 3ngenerated fundamental group. 1. introduction A Heegaard surface of an orientable closed 3manifold M is an embedded orientable surface S such that M − S consists of 2 handlebodies V1 and V2. This decomposition of M is called a Heegaard splitting and denoted by M = V1 ∪S V2. We say that the splitting is of genus g if S is of genus g. It is not difficult to see that any orientable closed 3manifold admits a Heegaard splitting. If M admits a Heegaard splitting of genus g but no Heegaard splitting of smaller genus then we say that M has Heegaard genus g and write g(M) = g. Clearly any curve in a handlebody can be homotoped to its boundary. It follows that for any Heegaard splitting M = V1 ∪S V2 every curve in M can be homotoped into V1. Thus the map induced by the inclusion of V1 into M maps a generating set of π1(V1) to a generating set of π1(M). As π1(V1) is generated by g elements it follows that π1(M) is also generated by g elements. Thus g(M) ≥ r(M) where
Topological Index Theory for surfaces in 3manifolds
 Geometry & Topology
"... The disk complex of a surface in a 3–manifold is used to define its topological index. Surfaces with welldefined topological index are shown to generalize well known classes, such as incompressible, strongly irreducible and critical surfaces. The main result is that one may always isotope a surface ..."
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Cited by 15 (6 self)
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The disk complex of a surface in a 3–manifold is used to define its topological index. Surfaces with welldefined topological index are shown to generalize well known classes, such as incompressible, strongly irreducible and critical surfaces. The main result is that one may always isotope a surface H with topological index n to meet an incompressible surface F so that the sum of the indices of the components of H nN.F / is at most n. This theorem and its corollaries generalize many known results about surfaces in 3–manifolds, and often provides more efficient proofs. The paper concludes with a list of questions and conjectures, including a natural generalization of Hempel’s distance to surfaces with topological index 2. 57M99 1
LENS SPACE SURGERIES & PRIMITIVE/TOROIDAL CONSTRUCTIONS
, 2008
"... We show that lens space surgeries on knots in S³ which arise from the primitive/toroidal construction also arise from the primitive/primitive construction. This continues a program to address the Berge Conjecture in the case of tunnel number one knots. ..."
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Cited by 1 (1 self)
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We show that lens space surgeries on knots in S³ which arise from the primitive/toroidal construction also arise from the primitive/primitive construction. This continues a program to address the Berge Conjecture in the case of tunnel number one knots.
STABILIZATIONS OF HEEGAARD SPLITTINGS OF GRAPH MANIFOLDS
, 2008
"... We show that after one stabilization, a strongly irreducible Heegaard splitting of suitably large genus of a graph manifold is isotopic to an amalgamation along a modified version of the system of canonical tori in the JSJ decomposition. As a corollary, two strongly irreducible Heegaard splittings ..."
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We show that after one stabilization, a strongly irreducible Heegaard splitting of suitably large genus of a graph manifold is isotopic to an amalgamation along a modified version of the system of canonical tori in the JSJ decomposition. As a corollary, two strongly irreducible Heegaard splittings of a graph manifold of suitably large genus are isotopic after at most one stabilization of the higher genus splitting.
ON NONSIMPLE KNOTS IN LENS SPACES WITH TUNNEL NUMBER ONE
, 2009
"... A knot k in a closed orientable 3manifold is called nonsimple if the exterior of k possesses a properly embedded essential surface of nonnegative Euler characteristic. We show that if k is a nonsimple prime tunnel number one knot in a lens space M (where M does not contain any embedded Klein bott ..."
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A knot k in a closed orientable 3manifold is called nonsimple if the exterior of k possesses a properly embedded essential surface of nonnegative Euler characteristic. We show that if k is a nonsimple prime tunnel number one knot in a lens space M (where M does not contain any embedded Klein bottles), then k is a (1,1) knot. Elements of the proof include handle addition and Dehn filling results/techniques of Jaco, EudaveMuñoz and Gordon as well as structure results of Schultens on the Heegaard splittings of graph manifolds.