  We study knots in $\mathbb{S}^3$ obtained by the intersection of a minimal
surface in $\mathbb{R}^4$ with a small 3-sphere centered at a branch point. We
construct examples of new minimal knots. In particular we show the existence of
non-fibered minimal knots. We show that simple minimal knots are either
reversible or fully amphicheiral; this yields an obstruction for a given knot
to be an iterated knot of a minimal surface. Properties and invariants of these
knots such as the algebraic crossing number of a braid representative and the
Alexander polynomial are studied.
