Stable functions and common stabilizations of Heegaard splittings

Johnson, Jesse

doi:10.1090/s0002-9947-09-04731-x

Cited by 18 publications

(18 citation statements)

References 24 publications

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“…Then either g splits f , in which case, by Lemma 33, By definition, if Σ splits S, then there are sweep-outs f and g representing S and Σ, respectively, such that f × g is generic and g splits f . As noted above, the boundaries of the closures of R a and R b are edges of the graphic for f × g. As pointed out in [8], a horizontal tangency in the graphic for f × g corresponds to a critical point in the function g. Since g is a sweep-out, it has no critical points away from its spines, and so there can be no horizontal tangencies in the interior of the graphic. Thus the maxima of the upper boundary ofR a and minima of the lower boundary ofR b are vertices of the graphic.…”

Section: Splitting Sweep-outsmentioning

confidence: 88%

Bounding the stable genera of Heegaard splittings from below

Johnson

2010

Journal of Topology

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We describe for each positive integer k a 3‐manifold with Heegaard surfaces of genus 2k and 2k−1 such that any common stabilization of these two surfaces has genus at least 3k−1. We also find, for every k, a 3‐manifold with boundary admitting Heegaard splittings of genus k and k+1 whose stable genus is 2k, and for every positive n, a 3‐manifold that has n pairwise nonisotopic Heegaard splittings of the same genus all of which are stabilized.

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Section: Splitting Sweep-outsmentioning

confidence: 88%

Bounding the stable genera of Heegaard splittings from below

Johnson

2010

Journal of Topology

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“…A region of the graphic is a component of the complement of the graphic in the unit square. We say that φ T × φ S is generic if it is stable [33] on the complement of the spines and each line {t} × [0, 1] and [0, 1] × {s} contains at most one vertex of the graphic. (Stable functions are those which, in the words of [33], have the property that "small perturbations do not change the topology".…”

Section: The Graphicmentioning

confidence: 99%

Exceptional and cosmetic surgeries on knots

et al. 2016

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We show that the bridge distance of a knot determines a lower bound on the genera of essential surfaces and Heegaard surfaces in the manifolds that result from non-trivial Dehn surgeries on the knot. In particular, knots with high bridge distance do not admit non-trivial non-hyperbolic surgeries or non-trivial cosmetic surgeries. We further show that if a knot has bridge distance at least 3 then its bridge number is bounded above by a function of Seifert genus, or indeed by the genus of (almost) any essential surface or Heegaard surface in the surgered manifold.

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“…Note that, the stable mappings in C ∞ (X, Y ) always form an open subset. Now, we make use of the following observation, which can be found, for example, in [9]. Assume that two stable mappings in C ∞ (X, Y ) are connected by a path γ consisting of stable mappings.…”

Section: Lemmamentioning

confidence: 99%

Equilibria of Riesz Potentials Generated by Point Charges at the Roots of Unity

Bilogliadov

2015

Comput. Methods Funct. Theory

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We consider a special case of Maxwell's problem on the number of equilibrium points of the Riesz potential 1/r 2β (where r is the Euclidean distance and β is the Riesz parameter) for positive unit point charges placed at the vertices of a regular polygon. We show that the equilibrium points are located on the perpendicular bisectors to the sides of the regular polygon, and study the asymptotic behavior of the equilibrium points with regard to the number of charges n and the Riesz parameter β. Finally, we prove that for values of β in a small left-hand side neighborhood of β = 1, the Riesz potential has only one equilibrium point different from the origin on each perpendicular bisector, and one equilibrium point at the origin.

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Stable functions and common stabilizations of Heegaard splittings

Cited by 18 publications

References 24 publications

Bounding the stable genera of Heegaard splittings from below

Bounding the stable genera of Heegaard splittings from below

Exceptional and cosmetic surgeries on knots

Equilibria of Riesz Potentials Generated by Point Charges at the Roots of Unity

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