I am interested in this question since 1999 when I heared the definition of a knot and I read the definition of unknoting and the total curvature of a knot.
To what extent can closed orbits of an arbitrary algebraic vector field of degree $n$ on $\mathbb{R}^3$ have complicated knot structure? More precisely are there uniform upper bound $K(n)$ for the total curvature and unknoting number of a closed orbit of a polynomial vector field $X=P_n(x,y,z)\partial_x+Q_n(x,y,z)\partial_y+R_n(x,y,z)\partial_z$ where $P_n,Q_n,R_n$ are arbitrary polynomials of degree $n$
