Inventiones mathematicae

, Volume 150, Issue 2, pp 257-286

First online:

On the minimum ropelength of knots and links

  • Jason CantarellaAffiliated withDepartment of Mathematics, University of Georgia, Athens, GA 30602, USA (e-mail:
  • , Robert B. KusnerAffiliated withDepartment of Mathematics, University of Massachusetts, Amherst, MA 01003, USA (e-mail:
  • , John M. SullivanAffiliated withDepartment of Mathematics, University of Illinois, Urbana, IL 61801, USA (e-mail:

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The ropelength of a knot is the quotient of its length by its thickness, the radius of the largest embedded normal tube around the knot. We prove existence and regularity for ropelength minimizers in any knot or link type; these are C 1,1 curves, but need not be smoother. We improve the lower bound for the ropelength of a nontrivial knot, and establish new ropelength bounds for small knots and links, including some which are sharp.