Tight knot constant

Description of constant

$C_{22a}$ is the largest constant for which one has an inequality

$L\geq C_{22a}C^{3/4}$

for all knots, where $L$ is the ropelength of a knot (or link) with crossing number $C$. The ropelength $L$ is the infimum over all embeddings of the knot (or link) of the ratio of the contour length of the knot to its thickness. The thickness is defined as the radius of the smallest circle that passes through any three points on the knot (where collinear points yield an infinite radius). Colloquially, the ropelength is the least amount of rope required to tie a specific knot in a rope of unit radius. See [CKS2002] for the full definition.

Known upper bounds

Upper bounds are typically found by finding a tight instance of a specific knot using gradient descent (usually a torus knot).

Bound	Reference	Comments
12.81	[SDKP1998]	$8_{19}$ knot
12.63	[ACPR2011]	$10_{124}$ knot
10.76	[KM2021]	$T(25,26)$ knot

Known lower bounds

Bound	Reference	Comments
0.418	[DE1998]	Based on lattice embeddings.
$\left(\frac{4\pi}{11}\right)^{3/4}\approx 1.105$	[BS1999]	Argument based on an “electromagnetic” knot energy. The 11 in the denominator can be replaced by 10.67, bringing the bound to 1.13, but it is always reported as 11.

Additional comments

There is also a convention where the ropelength is defined relative to a unit-diameter rope, but unit-radius is more common.
Conjectured lower bound of 3.22 [Klotz2025] based on Mobius energy argument.
For knots without asymptotically large crossing number $C$ (currently below 1850) a stronger bound exists [Diao2003].
Alternating knots have a linear lower bound, discussed in 22b.

References