Exceptional and cosmetic surgeries on knots

Blair, Ryan; Campisi, Marion; Johnson, Jesse; Taylor, Scott A.; Tomova, Maggy

doi:10.1007/s00208-016-1392-3

Cited by 7 publications

(12 citation statements)

References 53 publications

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“…The distance of a bridge surface bounds below the genus of certain essential surfaces in the knot exterior [2], while Theorem 1.1 and Theorem 1.2 demonstrate an analogous property for height. It is known that both diagrams with large height and bridge surfaces with large distance produce knots with no exceptional surgeries [3,7]. Additionally, both height and bridge distance give strong restrictions on the Heegaard surfaces for the knot exterior [8,15].…”

Section: Then One Of Two Conclusion Holdsmentioning

confidence: 99%

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Essential surfaces in highly twisted link complements

Blair

Futer

Tomova

2015

Algebr. Geom. Topol.

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Abstract. We prove that in the complement of a highly twisted link, all closed, essential, meridionally incompressible surfaces must have high genus. The genus bound is proportional to the number of crossings per twist region. A similar result holds for surfaces with meridional boundary: such a surface either has large negative Euler characteristic, or is an n-punctured sphere visible in the diagram.

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Section: Then One Of Two Conclusion Holdsmentioning

confidence: 99%

“…Suppose that an arc γ runs from a boundary face to an adjacent interior edge in a face σ, violating the second half of condition (3). Then γ has endpoints in adjacent edges of ∂σ, and we may assume without loss of generality that it is outermost in σ.…”

Section: The Polyhedral Decompositionmentioning

confidence: 99%

Essential surfaces in highly twisted link complements

Blair

Futer

Tomova

2015

Algebr. Geom. Topol.

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“…[8],Lemma 2). If B is a bridge surface which is not a sphere with four or fewer punctures, then d BD (B, L) ≤ d C (B, L) ≤ 2d BD (B, L).…”

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confidence: 96%

Hyperbolic manifolds containing high topological index surfaces

Campisi

Rathbun

2018

Pacific J. Math.

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If a graph is in bridge position in a 3-manifold so that the graph complement is irreducible and boundary irreducible, we generalize a result of Bachman and Schleimer to prove that the complexity of a surface properly embedded in the complement of the graph bounds the graph distance of the bridge surface. We use this result to construct, for any natural number n, a hyperbolic manifold containing a surface of topological index n.

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“…• knots of bridge number at least 6 and distance at least 3, by Blair, Campisi, Johnson, Taylor and Tomova in 2012 [1].…”

Section: Introductionmentioning

confidence: 99%

“…al. [1], proving the Cabling Conjecture for 5-bridge knots restricts remaining cases to knots with low distance.…”

Section: Introductionmentioning

confidence: 99%

A combinatorial approach to the Cabling Conjecture

Grove¹

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To my family, whose idea of a recreational activity is working on a Millenium problem: thank you for allowing space for and encouraging my curiosity, and teaching me that regardless of success, difficult things are worth attempting.ii ABSTRACT Dehn surgery and the notion of reducible manifolds are both important tools in the study of 3-manifolds. The Cabling Conjecture of Francisco González-Acuña and Hamish Short describes the purported circumstances under which Dehn surgery can produce a reducible manifold. This thesis extends the work of James Allen Hoffman, who proved the Cabling Conjecture for knots of bridge number up to four. Hoffman built upon the combinatorial machinery used by Cameron Gordon and John Luecke in their solution to the knot complement problem. The combinatorial approach starts with the graphs of intersection of a thin level sphere of the knot and the reducing sphere in the surgered manifold. Gordon and Luecke's proof then proceeds by induction on certain cycles. Hoffman provides more insight into the structure of the base case of the induction (i.e. in an innermost cycle or a graph containing no such cycles). Hoffman uses this structure in a case-by-case proof of the Cabling Conjecture for knots of bridge number up to four.We find trees with specific properties in the graph of intersection, and use them to prove the existence of structure which provides lower bounds on the number of the aforementioned innermost cycles. Our results combined with a recent lower bound on the number of vertices inside the innermost cycles succinctly prove the conjecture for bridge number up to five and suggests an approach to the conjecture for knots of higher bridge number. iv PUBLIC ABSTRACTThe Earth's curvature is invisible to us, with our feet planted firmly on the ground.Yet the Earth is round. By the very roundness of the Earth, by our successes at leaving its surface, and by our study of the vastness through which it travels, we are compelled to be curious about the nature of 3-dimensional space.Before we recognized the Earth as round, we first had to recognize that round things existed, and consider their properties. Such steps are also crucial in considering 3-dimensional spaces. A useful method for determining properties of a 3-dimensional space is to ascertain how precisely that space differs from normal 3-dimensional space, and to ask what types of differences lead to which kinds of properties.

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Exceptional and cosmetic surgeries on knots

Cited by 7 publications

References 53 publications

Essential surfaces in highly twisted link complements

Essential surfaces in highly twisted link complements

Hyperbolic manifolds containing high topological index surfaces

A combinatorial approach to the Cabling Conjecture

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