Ideal knots

Ideal Knots

Although all configurations of a given knot are topologically equivalent, there is an Ideal [1] form which minimises the length to diameter ratio Typically there are several local minima of L/D differing by less than 1% - so it can be difficult to distinguish which is the global minimum. There is nothing to prevent a knot having two minima with the same value of L/D and certainly some are very close.

The database of Ideal Knots to 10-crossings gives for each knot the best conformation I have found.
Each is represented by a Fourier series of vectors A[i], B[i], i=1..100, so that

X(t) = ∑_i A[i]*cos(i.t) + B[i]*sin(i.t)

The file has the following format:-

<AB Id="3:1:1" Conway="3" L="16.372861" D=" 1.000000">
  <Coeff I= 1 A=" 0.374743, 0.000000, 0.000000" B=" 0.000000, 0.374482, 0.000000" />
  <Coeff I= 2 A=" 0.938789,-0.600971,-0.000244" B="-0.601135,-0.938461,-0.002035" />
  <Coeff I= 3 A=" 0.000030, 0.000830,-0.283298" B=" 0.000718, 0.000974,-0.442135" />
  ...
</AB>


The Id is crossing number, number of strings (1) and identifier given by Alexander and Briggs - also used by Rolfsen (both versions of the Perko pair are given). Thus 5:1:2 is knot 5₂
The Conway notation is also given. The values of L and D are calculated from a 512 point approximation to this curve. Zero coefficients are omitted.

To aid comparison of conformations they are standardised to a unique form: -

• the parameter t is linear with string length
• the parameter zero has been rebased, t → t+t₀, & rotated so that A[1]=(a,0,0), B[1]=(0,b,0), a > b > 0
• make A[2].x > 0 by selecting t₀ or t₀+π
• make A[2].y > 0, if necessary, by reversing the string: t → −t (negate all Ay, Az & Bx)
• make A[2].z > 0, if necessary, by reflecting in the X-Y plane (negate all Az & Bz)
• rescale so D = 1

This standardisation is not guaranteed to work always - for example the trefoil given above has A[1].x suspiciously close to B[1].y - if they should be the same the standardisation fails. But this only affects a few knots.

The conformations were obtained using the shrink-on-no-overlap (SONO [1, chapter 2]) algorithm with 512 points. Starting from a stylised layout generated from the Conway notation, this was randomly rearranged before applying SONO. Repeated runs produce about seven different conformations (average for 10 crossing knots), often with very similar values of L/D. It would take several dozen runs for the randomising to find the global minimum for most knots - and even then the wrong one may have been chosen because of the limited accuracy in locating the minimum. Thus, after only a few runs (representing many days of computing), many of the results will only be one of the best local minima.

Compared to published results up to 9 crossings [1, chapter 1] many in this database are about 0.25% worse. This difference arises in the conversion from the point representation used during iteration to the Fourier version: partly because of only using 100 coefficients. Also iteration tends to leave nodes nestled equidistant from both ends of the nearest leash - measuring D by node-to-node distance produces an error of about 0.1%; conversion and standardisation of the Fourier form generally shifts the nodes arbitarily and removes this error.

Only a few knots are > 0.3% worse, with one knot (9₁₆ 3,3,2+) 0.5% worse; these are probably instances where Stasiak et al found tighter conformations than those given here.

Brian Gilbert
Email: =<A href="mailto:brian.gilbert@xtra.co.nz">brian.gilbert@xtra.co.nz</A>
Ref: [1] Ideal Knots, vol.19 of Series on Knots and Everything, ed: Stasiak, Katritch and Kauffman, World Scientific 1998.