Specific information about the knots and computations are given below. For those with a short attention span, here are the links to the computations on different classes of knots.
The data comes from applying the methods of the paper Chirality of crooked curves by Giovanni Dietler, Robert Kusner, Wöden Kusner, Eric Rawdon, and Piotr Szymczak to different (see below) sets of knot configurations.
Each of the configurations have been translated so that its center of reaction R is at the origin. The configurations are then rotated so that the eigenvectors of the QR matrix align with the coordinate axes. In particular, the eigenvector associated with the eigenvalue with the smallest absolute value is aligned with the x-axis, the eigenvector associated with the eigenvalue with the middle absolute value is aligned with the y-axis, and the eigenvector associated with the eigenvalue with the largest absolute value is aligned with the z-axis.
Tight knots are also known as ideal or ropelength-minimized (for background, see e.g. references 24 and 25 of the paper). The configurations shown here are polygonal approximations of the configurations and were tightened using the program Ridgerunner. The number of edges in the polygons is approximately eight times the ropelength (ranging from 400ish to 900ish), except for the trefoil, which has 2400 edges.
KnotPlot is a program for playing with mathematical knots and doing some serious knot theory. The program has a library of pleasant looking configurations. The number of edges ranges from 50ish to 150ish.
Ridgerunner can tighten configurations while enforcing rotational symmetries. We tightened (2,n) torus knots with 2-fold rotational symmetry enforced. We include all (2,n) torus knots with n being odd and running from 3 to 25. These configurations have 24n vertices.
Ridgerunner can tighten configurations while enforcing rotational symmetries. We tightened (p,q) torus knots with both p- and q-fold rotational symmetry enforced. We include all (p,q) torus knots through 10 crossings. These all have around 220 vertices and the number of vertices is divisible by both p and q.
Ridgerunner can tighten configurations while enforcing rotational symmetries. We tightened all twist knots with 2-fold rotational symmetry enforced. We include all twist knots through 10 crossings. These configurations have 200 vertices.
The figure-eight knot, 4.1, has 3D realizations with a certain rotation and reflection symmetry. More specifically, there exists a configuration K0, a plane P, and a line L perpendicular to P such that a rotation of 90 degrees about L and reflection over P takes K0 to itself. If this symmetry is applied twice, we obtain the 180 degree 2-fold rotational symmetry.
The tight figure-eight knot, 4.1, either has, or is very close to having, this symmetry. See the left image on this page where we are looking down the line L and the plane P is your computer monitor. Unfortunately, Ridgerunner cannot tighten with this full symmetry enforced. Instead we ran a 200-edge figure-eight knot with 2-fold rotational symmetry enforced and then averaged the vertices to force the full symmetry. So this configuration is pretty tight, but it is not the tightest.
Since we can further tighten this very symmetric version of the 4.1, with or without the 2-fold rotational symmetry enforced, we might conjecture that the tight/ideal/ropelength-minimizing configuration of the figure-eight knot does not have this rotation and reflection symmetry. With that said, it could be that the lack of symmetry is due to the Ridgerunner algorithm and/or due to numerical instability.
This material is based upon work supported by the National Science Foundation under Grant Nos. 1115722, 1418869, and 1720342 for E. Rawdon, by the University of Pittsburgh, Graz University of Technology, Vanderbilt University, Austrian Science Fund (FWF) Project 5503, and National Science Foundation Grant Nos. 1516400 and 1104102 for W. Kusner, by National Science Centre (Poland) under Grant 2015/19/D/ST8/03199 for P. Szymczak, and by National Science Foundation Grant Nos. 1440140, 1439786, and 1607611 for R. Kusner. G. Dietler, R. Kusner, E. Rawdon, and P. Szymczak were supported by the Newton Institute in 2012.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.