Commentarii Mathematici Helvetici

, Volume 71, Issue 1, pp 617–627

Unknotting tunnels in two-bridge knot and link complements


  • Colin C. Adams
    • Department of MathematicsWilliams College
  • Alan W. Reid
    • Department of MathematicsUniversity of Texas

DOI: 10.1007/BF02566439

Cite this article as:
Adams, C.C. & Reid, A.W. Commentarii Mathematici Helvetici (1996) 71: 617. doi:10.1007/BF02566439


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).

Copyright information

© Birkhäuser Verlag 1996