Almost alternating links

Adams, Colin; Brock, Jeffrey; Bugbee, John; Comar, Timothy D.; Faigin, Keith; Huston, Amy M.; Joseph, Anne; Pesikoff, David

doi:10.1016/0166-8641(92)90130-r

Cited by 80 publications

(101 citation statements)

References 9 publications

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“…A typical example of accidental surfaces is represented by meridionally compressible surfaces, that is, for each of which there is a compression disk in S 3 meeting K at just one point. It is known that for alternating knots ( [7]), almost alternating knots ( [2]), toroidally alternating knots ( [1]), 3-braid knots ( [6]), and Montesinos knots ( [10]), every closed incompressible surface in their complements is meridionally compressible, and hence accidental.…”

Section: Introductionmentioning

confidence: 99%

Accidental surfaces in knot complements

Ichihara

Ozawa

2000

J. Knot Theory Ramifications

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It is well known that for many knot classes in the 3-sphere, every closed incompressible surface in their complements contains an essential loop which is isotopic into the boundary of the knot exterior. In this paper, we investigate closed incompressible surfaces in knot complements with this property. We show that if a closed, incompressible, non-boundary-parallel surface in a knot complement has such loops, then they determine the unique slope on the boundary of the knot exterior. Moreover, if the slope is non-meridional, then such loops are mutually isotopic in the surface. As an application, a necessary and sufficient condition for knots to bound totally knotted Seifert surfaces is given.

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Section: Introductionmentioning

confidence: 99%

Accidental surfaces in knot complements

Ichihara

Ozawa

2000

J. Knot Theory Ramifications

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“…W. Menasco has shown that prime alternating knots are either hyperbolic or torus knots [12]. It has been generalized by C. Adams that prime almost alternating knots are either hyperbolic or torus knots [2]. It is known that no satellite knot is an almost alternating knot [9].…”

Section: Cl(h − Int(a)) Is An Annulus In H Moreover One Can Find Amentioning

confidence: 99%

“…In 1960s, W. Haken [4] 2 ; H) is said to be strongly irreducible. They showed that all splittings of a non-Haken manifold are either reducible or strongly irreducible.…”

Section: Introductionmentioning

confidence: 99%

The Disjoint Curve Property and Bridge Surfaces

Hong¹,

Kim²

2009

Bulletin of the Korean Mathematical Society

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“…The Menasco-Reid conjecture has been shown true for many other classes of knots, including almost alternating knots [2], Montesinos knots [13], toroidally alternating knots [1], 3-bridge and double torus knots [5] and knots of braid index 3 [8] and 4 [9]. For a knot in one of the above families, any closed essential surface in its complement has a topological feature which obstructs it from being even quasi-Fuchsian.…”

Section: Introductionmentioning

confidence: 99%