Polynomial knots

See also: https://home.adelphi.edu/~stemkoski/knotgallery/

Polynomial knot 3-1

The XML code for the example below can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder in the files D01a_Path_Poly31Knot.3D.xml (D01a-c).

Polynomial Knot. https://home.adelphi.edu/~stemkoski/knotgallery/poly3.html

double t2 = t * t;
double t3 = t2 * t;
double t4 = t3 * t;
double t5 = t4 * t;
var x = (t3 - 3 * t) / 10;
var y = (t4 - 4 * t2) / 10;
var z = (t5 / 5 - 2 * t) / 10;

\(\quad\) \(\quad\)

Polynomial knot 4-1

The XML code for the example below can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder in the files D02a_Path_Poly41Knot.3D.xml (D02a-c).

Figure-Eight (41) Polynomial Knot. https://home.adelphi.edu/~stemkoski/knotgallery/poly3.html

double t2 = t * t;
double t4 = t2 * t2;
var x = (2.0 / 5.0) * t * (t2 - 7) * (t2 - 10) / 10;
var y = (t4 - 13 * t2) / 10;
var z = (1.0 / 10.0) * t * (t2 - 4) * (t2 - 9) * (t2 - 12) / 10;

\(\quad\) \(\quad\)

Polynomial knot 5-1

The XML code for the example below can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder in the files D03a_Path_Poly51Knot.3D.xml (D03a-c).

Cinquefoil (51) Polynomial Knot. https://home.adelphi.edu/~stemkoski/knotgallery/poly3.html

double t2 = t * t;
double t3 = t2 * t;
double t4 = t3 * t;
double t5 = t4 * t;
double t6 = t5 * t;
double t7 = t6 * t;
var x = (1.0 / 5.0) * (t5 - 36 * t3 + 260 * t) / 10;
var y = (1.0 / 2.0) * (t4 - 24 * t2) / 10;
var z = (1.0 / 100.0) * (t7 - 31 * t5 + 164 * t3 - 560 * t) / 10;

\(\quad\) \(\quad\)

Polynomial knot 6-2

The XML code for the example below can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder in the files D04a_Path_Poly62Knot.3D.xml (D04a-c).

Six-Crossing (62) Polynomial Knot. https://home.adelphi.edu/~stemkoski/knotgallery/poly3.html

double t2 = t * t;
double t3 = t2 * t;
double t4 = t3 * t;
var x = (3.0 / 4.0) * t * (t2 - 4) * (t2 - 11) / 10;
var y = (t4 - 12 * t2) / 10;
var z = (1.0 / 200.0) * t * (t2 - 1) * (t2 - 3) * (t2 - 9) * (t2 - 10) * (t2 - 12) / 10;

\(\quad\) \(\quad\)

Polynomial knot 7-4

The XML code for the example below can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder in the files D05a_Path_Poly74Knot.3D.xml (D05a-c).

Seven-Crossing (74) Polynomial Knot. https://home.adelphi.edu/~stemkoski/knotgallery/poly3.html

double t2 = t * t;
double t3 = t2 * t;
double t4 = t3 * t;
double t5 = t4 * t;
var x = (4.0 / 5.0) * t * (t2 - 6) * (t2 - 12) / 10;
var y = t2 * (t2 - 7) * (t2 - 9) / 10;
var z = (1.0 / 200.0) * t * (t2 - 0.2) * (t2 - 1) * (t2 - 5) * (t2 - 6.5) * (t2 - 9) * (t2 - 10) / 10;

\(\quad\) \(\quad\)