knottedness

Knottedness, in mathematics, is the property of a closed curve in three-dimensional space being knotted, i.e., not deformable to a simple round circle without cutting. More formally, a knot is an embedding of a circle into three-dimensional Euclidean space (or the 3-sphere). Two knots are considered the same, up to ambient isotopy, if one can be continuously deformed into the other without cutting or passing through itself.

The simplest knot is the unknot, which is equivalent to a plain circle. Most knots cannot be untied; their knottedness is detected via projections called knot diagrams, which are two-dimensional pictures of crossings with over-under information. Reidemeister moves show when two diagrams represent the same knot.

Knottedness is studied using invariants that persist under ambient isotopy. Examples include the fundamental group of the knot complement and polynomial invariants such as the Alexander polynomial, the Jones polynomial, and invariants from quantum topology. These tools help distinguish different knot types and prove that certain knots are nontrivial.

Notable nontrivial knots include the trefoil knot and the figure-eight knot. Knottedness also appears outside pure mathematics: in biology and chemistry, DNA strands and synthetic polymers can become entangled, influencing function and manufacture.

Historically, knot theory emerged in the late 19th and early 20th centuries, with contributions from mathematicians such as Tait, Reidemeister, and Alexander. Knottedness is distinct from linking, which concerns multiple components rather than a single closed curve.