Knot theory

Somewhat analogous to the problem in the note above is the problem of classifying knots. The pictures below show some of the simplest distinct knots. But given presentations of two knots, no finite procedure is known that determines in general whether the knots are equivalent (or constructs a sequence of Reidemeister moves that transform one into the other). Quite probably this is in general undecidable, though since the 1920s a few polynomial invariants have been discovered—with recent ones being related to ideas from quantum field theory—that have allowed some progress to be made. (Even the problem of determining whether a knot specified by line segments is trivial is known to be NP-complete.)