2011arXiv (Cornell University)Open access

Seifert fibered surgeries with distinct primitive/Seifert positions

Mario Eudave-Muñoz, Katura Miyazaki, Kimihiko Motegi

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Abstract

We call a pair (K, m) of a knot K in the 3-sphere S^3 and an integer m a Seifert fibered surgery if m-surgery on K yields a Seifert fiber space. For most known Seifert fibered surgeries (K, m), K can be embedded in a genus 2 Heegaard surface of S^3 in a primitive/Seifert position, the concept introduced by Dean as a natural extension of primitive/primitive position defined by Berge. Recently Guntel has given an infinite family of Seifert fibered surgeries each of which has distinct primitive/Seifert positions. In this paper we give yet other infinite families of Seifert fibered surgeries with distinct primitive/Seifert positions from a different point of view. In particular, we can choose such Seifert surgeries (K, m) so that K is a hyperbolic knot whose complement S^3 - K has an arbitrarily large volume.

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We call a pair (K, m) of a knot K in the 3-sphere S^3 and an integer m a Seifert fibered surgery if m-surgery on K yields a Seifert fiber space. For most known Seifert fibered surgeries (K, m), K can be embedded in a genus 2 Heegaard surface of S^3 in a primitive/Seifert position, the concept introduced by Dean as a natural extension of primitive/primitive position defined by Berge. Recently Guntel has given an infinite family of Seifert fibered surgeries each of which has distinct primitive/Seifert positions. In this paper we give yet other infinite families of Seifert fibered surgeries with distinct primitive/Seifert positions from a different point of view. In particular, we can choose such Seifert surgeries (K, m) so that K is a hyperbolic knot whose complement S^3 - K has an arbitrarily large volume.

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Available abstract

We call a pair (K, m) of a knot K in the 3-sphere S^3 and an integer m a Seifert fibered surgery if m-surgery on K yields a Seifert fiber space. For most known Seifert fibered surgeries (K, m), K can be embedded in a genus 2 Heegaard surface of S^3 in a primitive/Seifert position, the concept introduced by Dean as a natural extension of primitive/primitive position defined by Berge. Recently Guntel has given an infinite family of Seifert fibered surgeries each of which has distinct primitive/Seifert positions. In this paper we give yet other infinite families of Seifert fibered surgeries with distinct primitive/Seifert positions from a different point of view. In particular, we can choose such Seifert surgeries (K, m) so that K is a hyperbolic knot whose complement S^3 - K has an arbitrarily large volume.

Key concepts: Fibered knot, Seifert surface, Mathematics, Knot (papermaking), Dehn surgery, Combinatorics, Genus, Anatomy

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