Tree-like curve

In mathematics, particularly in differential geometry, a tree-like curve is a generic immersion

with the property that removing any double point splits the curve into exactly two disjoint connected components.

They were first systematically studied by Russian mathematicians Boris Shapiro and Vladimir Arnold in the 1990s.

[1][2] For generic curves interpreted as the shadows of knots (that is, knot diagrams from which the over-under relations at each crossing have been erased), the tree-like curves can only be shadows of the unknot.

As knot diagrams, these represent connected sums of figure-eight curves.