THE POINCARÉ HOMOLOGY SPHERE, LENS SPACE SURGERIES, AND SOME KNOTS WITH TUNNEL NUMBER TWO

Kenneth L. Baker; Neil R. Hoffman

doi:10.2140/pjm.2020.305.1

THE POINCARÉ HOMOLOGY SPHERE, LENS SPACE SURGERIES, AND SOME KNOTS WITH TUNNEL NUMBER TWO

Kenneth L. Baker, Neil R. Hoffman

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We exhibit an infinite family of knots in the Poincare homology sphere with tunnel number 2 that have a lens space surgery. Notably, these knots are not doubly primitive and provide counterexamples to a few conjectures. Additionally, we update (and correct) our earlier work on Hedden’s almost-simple knots. In the appendix, it is shown that a hyperbolic knot in the Poincare homology sphere with a lens space surgery has either no symmetries or just a single strong involution.

Original language	English (US)
Pages (from-to)	1-27
Number of pages	27
Journal	Pacific Journal of Mathematics
Volume	305
Issue number	1
DOIs	https://doi.org/10.2140/pjm.2020.305.1
State	Published - Mar 2020

Keywords

Dehn surgery
lens space
Poincaré homology sphere
tunnel number

ASJC Scopus subject areas

Mathematics(all)

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10.2140/pjm.2020.305.1

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abstract = "We exhibit an infinite family of knots in the Poincare homology sphere with tunnel number 2 that have a lens space surgery. Notably, these knots are not doubly primitive and provide counterexamples to a few conjectures. Additionally, we update (and correct) our earlier work on Hedden{\textquoteright}s almost-simple knots. In the appendix, it is shown that a hyperbolic knot in the Poincare homology sphere with a lens space surgery has either no symmetries or just a single strong involution.",

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note = "Funding Information: Baker was partially supported by a grant from the Simons Foundation (#209184). Hoffman was supported by the Australian Research Council Discovery Grant DP130103694 and by a grant from the Simons Foundation (#524123). The appendix benefited from the thoughtful suggestions and corrections of the referee.",

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