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Knot & Link Invariants

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Abstractions about knots and links classified up to equivalence — polynomial invariants such as the bracket and HOMFLY polynomials, numerical invariants like the linking number, and structural classes such as torus knots, virtual knots, and link concordance, plus the related perfect spline construction.

9 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.

  • Bracket polynomial — A Laurent-polynomial state-sum invariant of framed unoriented link diagrams whose normalization yields the Jones polynomial for oriented links.
  • HOMFLY polynomial — A two-variable oriented-link invariant defined by a skein relation and normalization that specializes to the Alexander and Jones polynomials.
  • 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.
  • Perfect spline — A univariate spline of order m whose m-th derivative takes alternating values plus or minus one between successive knots.
  • 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.