Non-integer surgery and branched double covers of alternating knots

McCoy, Duncan

doi:10.1112/jlms/jdv030

Cited by 9 publications

(14 citation statements)

References 23 publications

(44 reference statements)

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“…We begin first by constructing a basis for a p/q-changemaker lattice. See also [McC15], [McC16]. Let…”

Section: Changemaker Latticesmentioning

confidence: 99%

“…When e ≥ 3 the Seifert fibered space is the branched double cover of an alternating Montesinos link. This allows us to apply previous results describing when the double branched cover of an alternating link can arise by non-integer surgery [McC15]. Although the results of [McC15] were derived using changemaker lattices, we do not explicitly use lattice theoretic techniques in this part of the proof.…”

Section: Introductionmentioning

confidence: 99%

“…This allows us to apply previous results describing when the double branched cover of an alternating link can arise by non-integer surgery [McC15]. Although the results of [McC15] were derived using changemaker lattices, we do not explicitly use lattice theoretic techniques in this part of the proof. We prove the theorem by considering Conway spheres in alternating diagrams of Montesinos links.…”

Section: Introductionmentioning

confidence: 99%

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The realization problem for non-integer Seifert fibered surgeries

Issa¹,

McCoy²

2018

Preprint

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Conjecturally, the only knots in S 3 with non-integer surgeries producing Seifert fibered spaces are torus knots and cables of torus knots. In this paper, we make progress on the associated realization problem. Let Y be a small Seifert fibered space bounding a positive definite plumbing with central vertex of weight e such that Y arises by non-integer p/q-surgery on a knot in S 3 . We show that if e ≥ 2 and the slope p/q is negative, or e ≥ 3 and p/q is positive, then Y can be obtained by p/q-surgery on a torus knot or a cable of a torus knot.

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“…We begin first by constructing a basis for a p/q-changemaker lattice. See also [McC15], [McC16]. Let…”

Section: Changemaker Latticesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The realization problem for non-integer Seifert fibered surgeries

Issa¹,

McCoy²

2018

Preprint

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“…We will establish some restrictions on changemaker lattice which admits an obtuse superbase. The following proposition, which combines results from [17] and [16], will allow us to restrict our attention to integer changemaker lattices.…”

Section: 2mentioning

confidence: 99%

“…By reflecting D, if necessary, we may assume that n is positive. It can be shown (e.g [16,Proposition 5.4]) there are tangle replacements showing that S 3 r (K) is an alternating surgery for all r in the range n ≤ r ≤ n + 1. Remark 5.1.…”

Section: 1mentioning

confidence: 99%

Bounds on alternating surgery slopes

McCoy

2017

Algebr. Geom. Topol.

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We show that if p/q-surgery on a nontrivial knot K yields the branched double cover of an alternating knot, then |p/q| ≤ 4g(K) + 3. This generalises a bound for lens space surgeries first established by Rasmussen. We also show that all surgery coefficients yielding the double branched covers of alternating knots must be contained in an interval of width two and this full range can be realised only if the knot is a cable knot. The work of Greene and Gibbons shows that if S 3 p/q (K) bounds a sharp 4-manifold X, then the intersection form of X takes the form of a changemaker lattice. We extend this to show that the intersection form is determined uniquely by the knot K, the slope p/q and the Betti number b 2 (X).

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Alternating knots with unknotting number one

McCoy

2017

Advances in Mathematics

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We prove that if an alternating knot has unknotting number one, then there exists an unknotting crossing in any alternating diagram. This is done by showing that the obstruction to unknotting number one developed by Greene in his work on alternating 3-braid knots is sufficient to identify all unknotting number one alternating knots. As a consequence, we also get a converse to the Montesinos trick: an alternating knot has unknotting number one if its branched double cover arises as half-integer surgery on a knot in S 3 . We also reprove a characterisation of almost-alternating diagrams of the unknot originally due to Tsukamoto.

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Non-integer surgery and branched double covers of alternating knots

Abstract: We show that if the branched double cover of an alternating link arises as p/q ∈ Q \ Z surgery on a knot in S 3 , then this is exhibited by a rational tangle replacement in an alternating diagram.

Cited by 9 publications

References 23 publications

The realization problem for non-integer Seifert fibered surgeries

The realization problem for non-integer Seifert fibered surgeries

Bounds on alternating surgery slopes

Alternating knots with unknotting number one

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