Reidemeister

Reidemeister moves are a foundational concept in knot theory, describing three local changes that can be applied to knot or link diagrams in the plane without altering the underlying knot type. They are named after Kurt Reidemeister, who introduced them in his 1926 work on knots and manifolds. The moves apply to planar projections of three-dimensional knots or links and are essential for understanding when two diagrams represent the same knot.

The three moves are as follows. Type I introduces or removes a twist in a single strand, creating or eliminating a single crossing and thereby changing the diagram’s writhe. Type II involves adding or removing two crossings by sliding one strand over another, essentially pairing two adjacent crossings that cancel each other. Type III moves a strand across a crossing between two other strands, reordering the crossings without changing their total number or the over/under relationships at those crossings.

A central result, known as the Reidemeister theorem, states that two oriented knot or link diagrams represent the same knot or link if and only if one can be transformed into the other by a finite sequence of these three moves (possibly along with planar isotopy). This theorem provides a combinatorial framework for knot equivalence and underpins many knot invariants, which are constructed to remain unchanged under the moves.

Reidemeister moves are fundamental tools in knot theory, used to analyze and simplify diagrams, prove equivalence, and study properties of knots and links. They also form a bridge between three-dimensional topology and planar diagrammatic representations.