## Commensurability classes of 2-bridge knot complements.(English)Zbl 1154.57001

The authors show that a hyperbolic 2-bridge knot complement is the unique knot complement in its commensurability class. They also discuss constructions of commensurable knot complements and conjecture that, for any hyperbolic knot $$K$$, there are at most three knot complements in the commensurability class of $$S^3 \setminus K$$. Furthermore, they conjecture that if $$K$$ additionally has no symmetries or hidden symmetries, then $$S^3 \setminus K$$ is the unique knot complement in its commensurability class.

### MSC:

 57M10 Covering spaces and low-dimensional topology 57M25 Knots and links in the $$3$$-sphere (MSC2010) 57M27 Invariants of knots and $$3$$-manifolds (MSC2010)

### Keywords:

commensurability; hyperbolic knot complement; 2-bridge knot
Full Text:

### References:

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