Instead of the unit sphere, consider the sphere of radius $\sqrt2$, containing the unit torus $\{ (z,w) : |z| = |w| = 1\}$. For each fixed value of $z$ in the torus, there are 5 solutions to the equation $z^2 + w^5 = 0$, arranged symmetrically about the circle in the $w$-coordinate. These are the fifth roots of $-z^2$. Similarly, for each value of $w$, there are two solutions to the equation (given by the square roots of $-w^5$) that are diametrically opposite in the unit $z$-circle. As you transport $z$ around its unit circle, the solutions to the equation will complete $2/5$ of a rotation in the $w$ coordinate. This yields a (2,5) torus knot. It is not hard to show that the equation has no other solutions in $S^3$.

I have no idea what was meant by the professor's remark. I usually think of knot theory as a subject defined by the objects one studies (namely knots), while algebraic topology is defined by the tools one uses. However, I don't hold this view with particularly strong conviction.