Dehn surgeries on knots which yield lens spaces and genera of knots

Okuno, Hiroshi G.; Teragaito, Masakazu

doi:10.1017/s0305004100004692

Cited by 28 publications

(83 citation statements)

References 36 publications

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“…In the blow-up B σ (R 1 ), each of these critical points gives rise to a collection of critical points a 0 i and a 1 i , corresponding to the eigenvalues λ i of the perturbed Dirac operator. We again assume these eigenvalues are labeled in increasing order, with λ 0 the first positive eigenvalue, as in (17) is in grading µ + 2i for i positive and in grading µ + 2i + 1 for i negative. The manifold N 1 has boundary R 1 , and its homology is generated by the classes [E 1 ] and [E 2 ] of the two spheres.…”

Section: 3mentioning

confidence: 99%

Monopoles and lens space surgeries

et al. 2007

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Monopole Floer homology is used to prove that real projective three-space cannot be obtained from Dehn surgery on a nontrivial knot in the three-sphere. To obtain this result, we use a surgery long exact sequence for monopole Floer homology, together with a nonvanishing theorem, which shows that monopole Floer homology detects the unknot. In addition, we apply these techniques to give information about knots which admit lens space surgeries, and to exhibit families of three-manifolds which do not admit taut foliations.

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Section: 3mentioning

confidence: 99%

Monopoles and lens space surgeries

et al. 2007

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“…By the Parity Rule, there are at most four edge classes in G T which we denote 1, α, β, αβ (see §4 [GT00] and [GL95]) as shown in Figure 10. Label an edge of G S by the class of the corresponding edge in G T .…”

Section: S ≥ 3 and T =mentioning

confidence: 99%

Once-punctured tori and knots in lens spaces

Baker

2011

Communications in Analysis and Geometry

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“…We adapt and build on some useful lemmas about order 2 and order 3 Scharlemann cycles and the faces of G S they bound from the work of Goda and Teragaito in [5]. They work with the case that s D r , but our generalization of this presents no problem here.…”

Section: Fundamental Lemmas About G Smentioning

confidence: 99%

Small genus knots in lens spaces have small bridge number

Baker

2006

Algebr. Geom. Topol.

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In a lens space X of order r a knot K representing an element of the fundamental group 1 X Š ‫=ޚ‬r ‫ޚ‬ of order s Ä r contains a connected orientable surface S properly embedded in its exterior X N .K/ such that @S intersects the meridian of K minimally s times. Assume S has just one boundary component. Let g be the minimal genus of such surfaces for K , and assume s 4g 1. Then with respect to the genus one Heegaard splitting of X , K has bridge number at most 1. 57M27; 57M25 Statement of resultsAny knot K in a lens space X D L.r; q/, r > 0, is rationally nullhomologous, ie OEK D 0 2 H 1 .X I ‫/ޑ‬ Š 0. We say r is the order of the lens space X , and we say the smallest positive integer s such that sOEK D 0 2 H 1 .X I ‫/ޚ‬ Š ‫=ޚ‬r ‫ޚ‬ is the order of the knot K . Note s Ä r . The exterior X N .K/ of K thus contains a connected properly embedded orientable surface S such that when S is oriented @S is coherently oriented on @ x N .K/ and intersects the meridian Â @ x N .K/ of K minimally s times, ie j @S j D s . Such a surface S is an analogue of a Seifert surface for a knot in S 3 . We refer to the genus of a knot K in X as the minimal genus of these "rational" Seifert surfaces for K . For this article we will restrict our attention to knots with rational Seifert surfaces that have just one boundary component.In this paper we prove the following theorem. Theorem 1.1 Let K be a genus g knot of order s in a lens space X whose Seifert surfaces have one boundary component. If s 4g 1 then, with respect to the Heegaard torus of X , K has bridge number at most 1. Theorem 1.1 may be curiously rephrased as saying small genus knots in lens spaces have small bridge number.In [1] Berge shows that double-primitive knots (ie simple closed curves that lie on a genus 2 Heegaard surface in S 3 and represent a generator of 1 for each handlebody)

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Dehn surgeries on knots which yield lens spaces and genera of knots

Cited by 28 publications

References 36 publications

Monopoles and lens space surgeries

Monopoles and lens space surgeries

Once-punctured tori and knots in lens spaces

Small genus knots in lens spaces have small bridge number

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