left click and move
rotating
scroll
dollying
right click and move
panning
double click
edit mode
Confinement Strength \(\mu/\kappa=\)

Model Properties:

Discretization Parameters:

Standard Knots:

03-Trefoil

(48 pts)04-Figure 8

(70 pts)05-1 Cinquefoil

(70 pts)05-2 Three-twist knot

(82 pts)06-1 Stevedore

(97 pts)06-2

(83 pts)06-3

(86 pts)07_001

(95 pts)07_002

(87 pts)07_003

(123 pts)07_004

(103 pts)07_005

(99 pts)07_006

(103 pts)07_007

(92 pts)08_001

(97 pts)08_002

(109 pts)08_003

(113 pts)08_004

(106 pts)08_005

(106 pts)08_006

(91 pts)08_007

(89 pts)08_008

(101 pts)08_009

(99 pts)08_010

(86 pts)08_011

(91 pts)08_012

(95 pts)08_013

(91 pts)08_014

(99 pts)08_015

(87 pts)08_016

(86 pts)08_017

(82 pts)08_018

(98 pts)08_019

(87 pts)08_020

(89 pts)08_021

(90 pts)09_001

(103 pts)09_002

(109 pts)09_003

(99 pts)09_004

(90 pts)09_005

(115 pts)09_006

(99 pts)09_007

(95 pts)09_008

(92 pts)09_009

(104 pts)09_010

(113 pts)09_011

(121 pts)09_012

(114 pts)09_013

(124 pts)09_014

(103 pts)09_015

(117 pts)09_016

(134 pts)09_017

(112 pts)09_018

(109 pts)09_019

(119 pts)09_020

(121 pts)09_021

(107 pts)09_022

(122 pts)09_023

(114 pts)09_024

(111 pts)09_025

(103 pts)09_026

(116 pts)09_027

(90 pts)09_028

(104 pts)09_029

(113 pts)09_030

(112 pts)09_031

(121 pts)09_032

(128 pts)09_033

(131 pts)09_034

(116 pts)09_035

(132 pts)09_036

(120 pts)09_037

(114 pts)09_038

(108 pts)09_039

(107 pts)09_040

(102 pts)09_041

(121 pts)09_042

(121 pts)09_043

(106 pts)09_044

(93 pts)09_045

(115 pts)09_046

(125 pts)09_047

(139 pts)09_048

(117 pts)09_049

(112 pts)10_001

(125 pts)10_002

(135 pts)10_003

(117 pts)10_004

(122 pts)10_005

(141 pts)10_006

(125 pts)10_007

(123 pts)10_008

(138 pts)10_009

(119 pts)10_010

(115 pts)10_011

(113 pts)10_012

(124 pts)10_013

(114 pts)10_014

(118 pts)10_015

(127 pts)10_016

(126 pts)10_017

(117 pts)10_018

(105 pts)10_019

(116 pts)10_020

(102 pts)10_021

(103 pts)10_022

(115 pts)10_023

(115 pts)10_024

(111 pts)10_025

(117 pts)10_026

(106 pts)10_027

(138 pts)10_028

(115 pts)10_029

(113 pts)10_030

(121 pts)10_031

(113 pts)10_032

(125 pts)10_033

(118 pts)10_034

(120 pts)10_035

(116 pts)10_036

(131 pts)10_037

(124 pts)10_038

(119 pts)10_039

(136 pts)10_040

(113 pts)10_041

(122 pts)10_042

(137 pts)10_043

(117 pts)10_044

(99 pts)10_045

(119 pts)10_046

(137 pts)10_047

(135 pts)10_048

(130 pts)10_049

(119 pts)10_050

(126 pts)10_051

(113 pts)10_052

(121 pts)10_053

(112 pts)10_054

(115 pts)10_055

(107 pts)10_056

(117 pts)10_057

(110 pts)10_058

(113 pts)10_059

(126 pts)10_060

(111 pts)10_061

(124 pts)10_062

(119 pts)10_063

(101 pts)10_064

(123 pts)10_065

(118 pts)10_066

(126 pts)10_067

(123 pts)10_068

(111 pts)10_069

(117 pts)10_070

(119 pts)10_071

(108 pts)10_072

(137 pts)10_073

(119 pts)10_074

(148 pts)10_075

(138 pts)10_076

(121 pts)10_077

(114 pts)10_078

(147 pts)10_079

(128 pts)10_080

(119 pts)10_081

(118 pts)10_082

(120 pts)10_083

(115 pts)10_084

(127 pts)10_085

(114 pts)10_086

(108 pts)10_087

(117 pts)10_088

(124 pts)10_089

(136 pts)10_090

(123 pts)10_091

(125 pts)10_092

(142 pts)10_093

(126 pts)10_094

(134 pts)10_095

(142 pts)10_096

(147 pts)10_097

(136 pts)10_098

(128 pts)10_099

(139 pts)10_100

(118 pts)10_101

(110 pts)10_102

(127 pts)10_103

(119 pts)10_104

(110 pts)10_105

(117 pts)10_106

(116 pts)10_107

(118 pts)10_108

(114 pts)10_109

(119 pts)10_110

(128 pts)10_111

(131 pts)10_112

(125 pts)10_113

(120 pts)10_114

(134 pts)10_115

(138 pts)10_116

(138 pts)10_117

(140 pts)10_118

(119 pts)10_119

(130 pts)10_120

(129 pts)10_121

(124 pts)10_122

(119 pts)10_123

(128 pts)10_124

(103 pts)10_125

(100 pts)10_126

(110 pts)10_127

(123 pts)10_128

(121 pts)10_129

(123 pts)10_130

(111 pts)10_131

(122 pts)10_132

(118 pts)10_133

(116 pts)10_134

(101 pts)10_135

(122 pts)10_136

(95 pts)10_137

(100 pts)10_138

(124 pts)10_139

(121 pts)10_140

(107 pts)10_141

(130 pts)10_142

(98 pts)10_143

(111 pts)10_144

(152 pts)10_145

(137 pts)10_146

(119 pts)10_147

(96 pts)10_148

(117 pts)10_149

(122 pts)10_150

(96 pts)10_151

(111 pts)10_152

(128 pts)10_153

(109 pts)10_154

(127 pts)10_155

(125 pts)10_156

(114 pts)10_157

(121 pts)10_158

(109 pts)10_159

(135 pts)10_160

(135 pts)10_161

(130 pts)10_162

(120 pts)10_163

(113 pts)10_164

(132 pts)10_165

(138 pts)10_166

(139 pts)
Or Enter a Parametric Representation for \(x\in[0,1]\)
A re-parametrization to arc-length is applied automatically.
\(y_1(x)=\)
\(y_2(x)=\)
\(y_3(x)=\)
Number of Discrete Points

Boundary Conditions:

Or Enter a Torus Knot
First Degree \(p=\)
Second Degree \(q=\)
Number of Discrete Points

Draw a Braid Representation:
Erase
Strand Distance per Diameter:
Distance at Crossings per Diameter:

Enter JSON data:

Confining Shape:
None

Ellipsoid

Cylinder along \(x\)

Cylinder along \(y\)

Cylinder along \(z\)

Box

Confinement Semi-Axes:
\(R_x=\)
\(R_y=\)
\(R_z=\)
For no confinement, enter "0" along the corresponding direction.

Confinement Center of Mass:
\(m_x=\)
\(m_y=\)
\(m_z=\)

Visible Objects:
Axes
Tube
Spheres
Tangents
Curvature
Confinement
Colorscale

Color Scheme:
Exponential: purple to yellow
Linear: red to yellow
Quadratic: grey to red

Curve Visualization
Degree of Interpolation:
Curve Thickness:

Energy Warning
Relative energy increase that triggers a warning: 10^

Data of Current Step

Downloads for Current Step

Downloads for Total Evolution:

Video of Canvas Evolution

Welcome to KNOTEVOLVE!

You are watching a "" curve with discrete points. The total length is .

If you would like to load a different knot or to change the elastic parameters, click on the "Parameter" button. You may choose from a set of predefined knots (all closed). As an alternative, a torus knot, which is defined by two winding-numbers, can be calculated. Also, loading a knot based on braids is possible. Based on a list of points or an explicit equation, rods with other types of boundary conditions can be calculated as well. All curves are automatically reparametrized to arc-length. A list of the current parameters and the curve position can be obtained in the "Export" tab in the section of "Data of Current Step".

To confine the rod/knot to a set of predefined shapes, please click on the "Confinement" tab. We offer spheres, cylinders and boxes.

The knot visualization can be rotated, zoomed and panned. The colors and other render options can be modified in the "Rendering" tab. If you double click on the figure, the knot becomes editable and you can move points around, coarsen, and refine the existing knot. On mobile devices, some features might not be available.

What's happening

Given a curve \(u:I\to\mathbf{R}^{3}\) for an interval \(I=(0,L)\) that is parametrized by arc-length (i.e. \(|u'|=1\)), we define the energy as the functional \[E[u]=\frac\kappa2\int_I\left|u''(x)\right|^2\mathrm{d}x + \varrho\,\mathrm{TP}[u] + \mu\,\mathrm{VP}[u],\] where \(\kappa\) is the bending rigidity, \(\varrho\) the self-repulsion, and \(\mu\) the confinement strength. The energy includes the curvature term \(\left|u''(x)\right|^2\), the tangent-point functional \[\mathrm{TP}[u]=\frac{2^{-2\ell}}{2\ell}\iint_{I\times I}\frac1{\mathbf r^{2\ell}(u(y), u(x))}\mathrm{d}x\mathrm{d}y\] for a suitable \(\ell>1\) (default: \(\ell=1.5\)) and the the domain violation penalty \[\mathrm{VP}[u]=\int_I\left(\sqrt{(u(x)-m_D)^T\,Q_D\,(u(x)-m_D)}-1\right)_+^2\] for a positive semi-definite quadratic form \(Q_D\) and the domain center of mass \(m_D\). The temporal evolution is calculated as a gradient flow \[(\partial_t u,w)+h^r([\partial_t u]'',w'')=-\delta E[u][w]\] for all admissible test curves \(w\) and the L2 inner product \((\cdot,\cdot)\), where \(h^r=0\) if \(r<0\).

The implemented scheme has an energy monotonicity property with controlled arc-length constraint violation for suitably chosen parameters. The algorithm is semi-implicit and based on cubic finite elements to represent the rod.

A detailed description of the calculations performed by KNOTEVOLVE is given in this manual.

About this Page

This is KNOTEVOLVE v20.12.
This software is under development and may change in future releases.
The service may be unavailable or under maintenance at any time.

Contributors: Sören Bartels, Philipp Falk, Pascal Weyer.
Further details can be found at the group webpage.
Please notify us in case you find any sort of bug on this page.

The authors thank Philipp Reiter and Patrick Schön for their contributions and insightful discussions as well as the Institute of Mathematics of the University of Freiburg for support and hospitality.
We only use technically necessary cookies that enable the communication between our server and the user's web browser.

We use and appreciate MathJax, THREE.js, Plotly, math.js, Font Awesome and Fira Sans. You may find copies of the respective licenses in this text file.

Literature

Sören Bartels, Philipp Reiter, and Johannes Riege. A simple scheme for the approximation of self-avoiding inextensible curves. IMA J. Numer. Anal., 38(2):543--565, 2018.

Sören Bartels. A simple scheme for the approximation of the elastic flow of inextensible curves. IMA J. Numer. Anal., 33(4):1115--1125, 2013.

O. Gonzalez and J. H. Maddocks. Global curvature, thickness, and the ideal shapesof knots. Proc. Natl. Acad. Sci. USA, 96(9):4769–4773, 1999.