MINIMISING THE BOUNDARIES OF PUNCTURED PROJECTIVE PLANES IN <font>S</font><sup>3</sup>

Domergue, Michel; Matignon, Daniel

doi:10.1142/s0218216501000937

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2003

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“…We know the RP 3 -conjecture to be satisfied for cable knots [42,43]. Furthermore, the standard spine of RP 3 is a projective plane and by [11], we know s ≥ 5.…”

Section: Introductionmentioning

confidence: 99%

Thin presentation of knots and lens spaces

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Matignon

2003

Algebr. Geom. Topol.

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This paper concerns thin presentations of knots K in closed 3-manifolds M 3 which produce S 3 by Dehn surgery, for some slope γ . If M does not have a lens space as a connected summand, we first prove that all such thin presentations, with respect to any spine of M have only local maxima. If M is a lens space and K has an essential thin presentation with respect to a given standard spine (of lens space M ) with only local maxima, then we show that K is a 0-bridge or 1-bridge braid in M ; furthermore, we prove the minimal intersection between K and such spines to be at least three, and finally, if the core of the surgery K γ yields S 3 by r -Dehn surgery, then we prove the following inequality: |r| ≤ 2g, where g is the genus of K γ .

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“…We know the RP 3 -conjecture to be satisfied for cable knots [42,43]. Furthermore, the standard spine of RP 3 is a projective plane and by [11], we know s ≥ 5.…”

Section: Introductionmentioning

confidence: 99%