Uniqueness of bridge surfaces for 2-bridge knots

Scharlemann, Martin; Tomova, Maggy

doi:10.1017/s0305004107000977

Cited by 29 publications

(44 citation statements)

References 15 publications

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“…In this case K is said to be removable as Q is also a Heegaard surface for M K after an isotopy of K . Scharlemann and Tomova discuss all four of these operations in detail in [11].…”

Section: Resultsmentioning

confidence: 99%

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Multiple bridge surfaces restrict knot distance

Tomova

2007

Algebr. Geom. Topol.

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Suppose M is a closed irreducible orientable 3-manifold, K is a knot in M , P and Q are bridge surfaces for K and K is not removable with respect to Q. We show that either Q is equivalent to P or d.K; P / Ä 2 .Q K/. If K is not a 2-bridge knot, then the result holds even if K is removable with respect to Q. As a corollary we show that if a knot in S 3 has high distance with respect to some bridge sphere and low bridge number, then the knot has a unique minimal bridge position.57M25, 57M27, 57M50

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“…In this case K is said to be removable as Q is also a Heegaard surface for M K after an isotopy of K . Scharlemann and Tomova discuss all four of these operations in detail in [11].…”

Section: Resultsmentioning

confidence: 99%

“…However 2 bridge knots in S 3 cannot have multiple bridge surfaces, Scharlemann and Tomova [11], so these cases don't arise in our context.…”

Section: The Curve Complex and Distance Of A Knotmentioning

confidence: 90%

Multiple bridge surfaces restrict knot distance

Tomova

2007

Algebr. Geom. Topol.

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“…These operations are discussed in detail in [7]. We give a brief overview of how sweepouts can be applied to study bridge surfaces for tangles in a 3-manifold.…”

Section: Obtaining New Bridge Splittings From Known Onesmentioning

confidence: 99%

“…These operations are discussed in detail by Scharlemann and Tomova [7] and they behave in a manner similar to stabilizations of Heegaard splittings. In this paper we consider pairs of bridge splittings † and † 0 for .M; T / and study bridge splittings † 00 that can be obtained from both † and † 0 via stabilizations and perturbations.…”

Section: Introductionmentioning

confidence: 98%

Flipping bridge surfaces and bounds on the stable bridge number

Johnson

Tomova

2011

Algebr. Geom. Topol.

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We show that if K is a knot in S 3 and † is a bridge sphere for K with high distance and 2n punctures, the number of perturbations of K required to interchange the two balls bounded by † via an isotopy is n. We also construct a knot with two different bridge spheres with 2n and 2n 1 bridges respectively for which any common perturbation has at least 3n 4 bridges. We generalize both of these results to bridge surfaces for knots in any 3-manifold.57M25, 57M27, 57M50

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“…Definition ( [17]). Let K be a knot in a closed, orientable 3-manifold M and let S be a Heegaard surface for M .…”

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confidence: 99%