Thin presentation of knots and lens spaces

This paper concerns thin presentations of knots K in closed 3-manifolds M 3 which produce S 3 by Dehn surgery, for some slope .I f Mdoes not have a lens space as a connected summand, we rst 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 nally, if the core of the surgery K yields S 3 by r -Dehn surgery, then we prove the following inequality: jrj 2 g ,w here gis the genus of K. AMS Classication 57M25; 57N10, 57M15

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