Parameterized Knots
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by Lee Stemkoski (Adelphi University)
Gallery I
Figure 8. Trefoil (31) Polynomial Knot
\[
\begin{eqnarray}
x(t) &=& t^3 - 3t \\
y(t) &=& t^4 - 4t^2 \\
z(t) &=& \frac15 t^5 - 2t \\
\end{eqnarray}
\]
Figure 9. Figure-Eight (41) Polynomial Knot
\[
\begin{eqnarray}
x(t) &=& \frac25 t (t^2 - 7)(t^2 - 10) \\
y(t) &=& t^4 - 13t^2 \\
z(t) &=& \frac1{10} t (t^2-4)(t^2-9)(t^2-12) \\
\end{eqnarray}
\]
Figure 10. Cinquefoil (51) Polynomial Knot
\[
\begin{eqnarray}
x(t) &=& \frac15 (t^5-36t^3+260t) \\
y(t) &=& \frac12 (t^4 - 24t^2) \\
z(t) &=& \frac1{100} (t^7+164t^3+560t) \\
\end{eqnarray}
\]
Figure 11. Six-Crossing (62) Polynomial Knot
\[
\begin{eqnarray}
x(t) &=& \frac34 t (t^2-4) (t^2-11) \\
y(t) &=& t^4 - 12t^2 \\
z(t) &=& \frac1{200} t(t^2-1)(t^2-3)(t^2-9)(t^2-10)(t^2-12)\\
\end{eqnarray}
\]
Figure 12. Seven-Crossing (74) Polynomial Knot
\[
\begin{eqnarray}
x(t) &=& \frac45 t (t^2-6)( (t^2-12) \\
y(t) &=& t^2 (t^2-7) (t^2-9) \\
z(t) &=& \frac1{200} t(t^2-0.2)(t^2-1)(t^2-5)(t^2-6.5)(t^2-9)(t^2-10)\\
\end{eqnarray}
\]
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