Distance of Heegaard splittings of knot complements

Tomova, Maggy

doi:10.2140/pjm.2008.236.119

Cited by 8 publications

(22 citation statements)

References 12 publications

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“…Thus K can be put in bridge position with respect to F so that 2 − χ( F − K) = 2 − (−2 − 2) = 6. Since d(S) > 6 by assumption, the main result of [41] implies that, in K ′ (γ), F is isotopic to a stabilization of S. Hence S has genus 2 (and F is isotopic to S in K ′ (γ)).…”

Section: Introductionmentioning

confidence: 93%

“…Note that 41-edge of h cannot lie in δ by Lemma 8.15. Since K is isotopic to the union of arc (12), two arcs on F , and one of the arcs (23) or (41), it is at most 1-bridge. Since the initial MST forms a type I FESC, we have a parallelism δ on G F between the leftmost edge of the M and the rightmost edge of the T. We may use this to attach the M to the S of the subsequent ST to form a FESC.…”

Section: Type I and Ii Fescsmentioning

confidence: 99%

“…Since γ 23 ⊂ ∂Θ 23 , it is either trivial on ∂T 23 or it runs three times longitudinally around T 23 . Furthermore, observe that (23) and (41) have bridge disks disjoint from D and A 23 ∪ Θ 23 and A 41 .…”

Section: 2mentioning

confidence: 99%

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Obtaining genus 2 Heegaard splittings from Dehn surgery

Baker

Gordon

Luecke

2013

Algebr. Geom. Topol.

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Let K' be a hyperbolic knot in S^3 and suppose that some Dehn surgery on K' with distance at least 3 from the meridian yields a 3-manifold M of Heegaard genus 2. We show that if M does not contain an embedded Dyck's surface (the closed non-orientable surface of Euler characteristic -1), then the knot dual to the surgery is either 0-bridge or 1-bridge with respect to a genus 2 Heegaard splitting of M. In the case M does contain an embedded Dyck's surface, we obtain similar results. As a corollary, if M does not contain an incompressible genus 2 surface, then the tunnel number of K' is at most 2.Comment: 127 pages, 81 figure

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Section: Introductionmentioning

confidence: 93%

Section: Type I and Ii Fescsmentioning

confidence: 99%

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Obtaining genus 2 Heegaard splittings from Dehn surgery

Baker

Gordon

Luecke

2013

Algebr. Geom. Topol.

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“…In order to prove Theorem 4.2 we use a generalization by Tomova [22] of a theorem of hers [23], which is itself a considerable refinement of a theorem of Scharlemann and Tomova [20] (see also Kobayashi and Rieck [14] for another proof).…”

Section: Ruling Out Simple Decompositionsmentioning

confidence: 99%

High distance knots

Minsky

Moriah

Schleimer

2007

Algebr. Geom. Topol.

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“…In the context of bridge splittings of knots, Bachman and Schleimer have used the arc and curve complex to adapt Hempel's distance to bridge splittings of knots, proving a result similar to that of Hartshorn: the distance of any splitting surface is bounded above by a function of the χ(S), where S is an essential surface in the knot exterior E(K) (see [1]). Further, Tomova [24] has proved that a distance similar to that of Bachman and Schleimer gives a lower bound on a function of χ(Σ ), where Σ is an alternate bridge surface for K .…”

Section: Introductionmentioning

confidence: 98%

Bridge and pants complexities of knots

Zupan

2012

Journal of the London Mathematical Society

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We modify an approach of Johnson to define the distance of a bridge splitting of a knot K in a 3-manifold M using the dual curve complex and pants complex of the bridge surface. This distance can be used to determine a complexity, which becomes constant after a sufficient number of stabilizations and perturbations, yielding an invariant of (M, K). We also give evidence toward the relationship between the pants distance of a bridge splitting and the hyperbolic volume of the exterior of K. Mathematics Subject Classification 57M25, 57M27, 57M50 (primary). 44 ALEXANDER ZUPAN Theorem 1.1. Let K be a knot in a closed, orientable 3-manifold M, where M has no S 2 × S 1 summands, let Σ be a splitting surface for (M, K), and let Σ h c be an (h, c)-stabilization of Σ. Then the limits lim h,c→∞ B(Σ) and lim h,c→∞ B P (Σ) exist. Moreover, they do not depend on Σ and thus define invariants B(M, K) and B P (M, K) of the pair (M, K).The pants complex P (Σ) of a surface Σ is itself an interesting object of study; for instance, Brock has shown that it is quasi-isometric to the Teichmüller space T (Σ) equipped with the Weil-Petersson metric [2], and Souto has revealed other surprising connections between pants distance and the geometry of certain 3-manifolds [23]. We exploit a theorem of Guéritaud and Futer [3] to prove the following theorem.Theorem 1.2. Suppose that K is a hyperbolic 2-bridge knot, Σ is a (0, 2)-splitting surface for K, v 3 is the volume of a regular ideal tetrahedron, and v 8 is the volume of a regular ideal octahedron. ThenThe paper proceeds in the following manner: in Section 2, we provide background information and include a proof of the analog of the Reidemeister-Singer theorem for bridge splittings in arbitrary manifolds. In Section 3, we introduce the curve, dual-curve, and pants complexes, and in Section 4 we define the distance of a bridge splitting and prove several basic facts about this distance. In Section 5, we use the distance of the previous section to define bridge and pants complexity, and prove the main theorem. In Section 6, we demonstrate several properties of these new invariants and, in Section 7, we define the concept of a critical splitting to provide explicit calculations in Section 8. Finally, in Section 9, we discuss connections between pants distance and hyperbolic volume, and in Section 10 we include several interesting open questions.

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Distance of Heegaard splittings of knot complements

Cited by 8 publications

References 12 publications

Obtaining genus 2 Heegaard splittings from Dehn surgery

Obtaining genus 2 Heegaard splittings from Dehn surgery

High distance knots

Bridge and pants complexities of knots

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