Knot, Link & Concordance Theory

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Abstractions about knots and links, invariants, linking number, Seifert surfaces, concordance, torus and virtual knots, and related continua.

8 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Knot invariant — A quantity, algebraic object or property assigned to a knot that is unchanged under the chosen knot-equivalence relation and can distinguish some inequivalent knots.
• Link (knot theory) — A finite disjoint union of smoothly or tamely embedded circles in three-dimensional space, considered up to ambient isotopy, with a knot as the one-component case.
• Link concordance — An equivalence between links whose components cobound disjoint embedded cylinders in one higher-dimensional spacetime.
• Linking number — An oriented integer invariant measuring how many times one disjoint closed curve winds around another in three-dimensional space.
• Seifert Surface — Span an oriented knot or link by a compact connected orientable embedded surface, turning a one-dimensional embedding into a two-dimensional carrier from which genus, linking forms, and knot invariants can be derived.
• Torus knot — A knot isotopic to a closed curve winding p and q times around the two generating directions of an unknotted torus, with coprime p and q.
• Virtual knot — An equivalence class of knot diagrams with classical and virtual crossings under classical Reidemeister moves and virtual detour moves, equivalently knots in thickened surfaces up to stabilization.
• Waraszkiewicz spiral — A member of an uncountable family of planar continua constructed so distinct members are incomparable under continuous surjections.