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Inspired by these two posts on knots orbits of polynomial vector fields on $\mathbb{R}^3$(A polynomial vector field on $\mathbb{R}^3$ which has a knot periodic orbit) and (Are total curvature and the unknoting number of closed orbits of algebraic vector fields bounded uniformly by the degree of vector field?) a similar question is given for links:

Are there uniform upper bounds $L(n)$ for the linking number of closed orbits of polynomial vector fields of degree $n$ on $\mathbb{R}^3$?

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The Lorenz equations are quadratic, and already have an infinite number of distinct knotted and linked orbits. An answer to one of your other questions has good references on Lorenz orbits, but I also like this nice set of slides from Joan Birman. To be concrete, torus knots / links of all types appear as orbits. This means you can get arbitrarily high linking numbers with a degree-two polynomial vector field.

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