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Full-Text Articles in Physical Sciences and Mathematics
On The Minimum Ropelength Of Knots And Links, Jason Cantarella, Robert B. Kusner, John M. Sullivan
On The Minimum Ropelength Of Knots And Links, Jason Cantarella, Robert B. Kusner, John M. Sullivan
Robert Kusner
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.
On Thickness And Packing Density For Knots And Links, Robert Kusner
On Thickness And Packing Density For Knots And Links, Robert Kusner
Robert Kusner
We describe some problems, observations, and conjectures concerning density of the hexagonal packing of unit disks in R2.thickness and packing density of knots and links in S3 and R3. We prove the thickness of a nontrivial knot or link in S3 is no more than 4 , the thickness of a Hopf link. We also give arguments and evidence supporting the conjecture that the packing density of thick links in R3 or S3 is generally less than √12 , the density of the hexagonal packing of unit disks in R2.