volume Menasco, applying Thurston's hyperbolization theorem for Haken manifolds, showed that any prime, non-split alternating link is hyperbolic, i.e. the link complement has a hyperbolic geometry, unless the link is a torus link. Thus hyperbolic volume is an invariant of many alternating links. Marc Lackenby has shown that the volume has upper and lower linear bounds as functions of the number of twist regions of a reduced, alternating diagram. References Further reading External links Celtic Knotwork to build an alternating knot from its planar graph Knot invariants