Knot theory

Knot theory studies embeddings of circles in three-dimensional space up to continuous deformation without cutting or passing strands through one another. It uses algebraic and geometric invariants to distinguish knots and links.

Ambient isotopy: A continuous deformation of the surrounding space that carries one embedded object to another without cutting or self-intersection. It formalizes when two embeddings represent the same knot.
Alexander polynomial: A Laurent polynomial invariant of knots and links, introduced by James W. Alexander in 1928. It is a computable invariant that detects some distinctions between knots.
Trefoil knot: The simplest nontrivial knot, with a diagram requiring at least three crossings. Its two mirror forms provide the first familiar example of chirality in knots.
Peter Guthrie Tait: A Scottish physicist and mathematician who compiled early knot tables in the nineteenth century. His knot tabulations helped establish knot classification as a mathematical project.
Knot diagram: A plane projection of a knot marked at each crossing to show which strand passes over the other. Diagrams turn three-dimensional knots into objects that can be drawn and manipulated.
Jones polynomial: A Laurent polynomial invariant of knots and links discovered by Vaughan Jones in 1984. It distinguishes examples that the Alexander polynomial cannot separate.
Figure-eight knot: The simplest knot that is not a trefoil, with minimal crossing number four. It is a compact example of a hyperbolic knot with unusually symmetric geometry.
James W. Alexander: An American mathematician whose work in topology included the Alexander polynomial. His polynomial provided one of the first powerful algebraic knot invariants.
Reidemeister moves: Three local changes to knot diagrams that preserve the represented knot up to ambient isotopy. They give a complete test for whether two diagrams depict equivalent knots.
Fundamental group: A group that records the homotopy classes of loops in a topological space. The fundamental group of a knot complement captures information about how the knot sits in space.

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