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Unknotting tunnels in two-bridge knot and link complements

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Commentarii Mathematici Helvetici

Abstract

We give a complete classification of the unknotting tunnels in 2-bridge link complements, proving that only the upper and lower tunnels are unknotting tunnels. Moreover, we show that the only strongly parabolic tunnels in 2-cusped hyperbolic 3-manifolds are exactly the upper and lower tunnels in 2-bridge knot and link complements. From this, it follows that the upper and lower tunnels in 2-bridge knot and link complements must be isotopic to geodesics of length at most ln(4), where length is measured relative to maximal cusps. Moreover, the four dual unknotting tunnels in a 2-bridge knot complement, which together with the upper and lower tunnels form the set of all known unknotting tunnels for these knots, must each be homotopic to a geodesic of length at most 6ln(2).

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Authors and Affiliations

  1. Department of Mathematics, Williams College, 01267, Williamstown, MA, USA

    Colin C. Adams

  2. Department of Mathematics, University of Texas, 78712, Austin, TX, USA

    Alan W. Reid

Authors
  1. Colin C. Adams
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  2. Alan W. Reid
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Additional information

First author supported by NSF Grant DMS-93028943, second author supported by the Royal Society.

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Adams, C.C., Reid, A.W. Unknotting tunnels in two-bridge knot and link complements. Commentarii Mathematici Helvetici 71, 617–627 (1996). https://doi.org/10.1007/BF02566439

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  • DOI: https://doi.org/10.1007/BF02566439

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