The ropelength of a smooth embedded closed space curve is its arc length divided by its thickness, where thickness is the supremum of the radii of embedded normal tubes about the curve (Cantarella et al. 2002). For a knot, the same term denotes the infimum over its representing curves. The ratio is unchanged by uniform rescaling. Thickness 1 uses a tube of radius 1, rather than diameter 1.
Klotz (2026) reports the lower bounds
 |
and
 |
where 
is ropelength, 
is the knot crossing number,
and 
is the stick
number. The first inequality improves the coefficient 1.105 in the earlier universal
crossing-number bound. Klotz (2026) describes mathematical assistance from ChatGPT-5.6
Luna in developing these bounds.
