Knot Invariants & Diagrammatic Algebra

← Back to Domain-Specific Families

Abstractions about brackets, knot polynomials, braid groups, graphical notation, crease or weave patterns, splines, and differential forms.

11 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.

• Basketweave (knitting) — Generate a woven-looking knitted texture by repeating offset blocks of knit-facing and purl-facing fabric, with block dimensions and alternation defining a recognizable stitch-pattern family.
• Bracket algebra — A quotient algebra encoding projective invariants through bracket symbols over a signed alphabet.
• Bracket polynomial — A Laurent-polynomial state-sum invariant of framed unoriented link diagrams whose normalization yields the Jones polynomial for oriented links.
• Crease pattern — A planar origami representation that records most or all folds of a finished model in one geometric diagram.
• Double affine braid group — A braid-like group associated with an affine root system that adds a second affine translation structure and whose group algebra leads to double affine Hecke algebras.
• HOMFLY polynomial — A two-variable oriented-link invariant defined by a skein relation and normalization that specializes to the Alexander and Jones polynomials.
• Murnaghan–Nakayama rule — A signed rim-hook removal rule for computing irreducible character values of symmetric groups from partitions.
• Nijenhuis–Richardson bracket — A graded Lie bracket on alternating vector-valued multilinear forms, defined by antisymmetrized insertion and used to encode Lie algebra structures and their deformations.
• Penrose graphical notation — A diagrammatic tensor notation in which shapes, lines and contractions visually encode multilinear maps, indices and composition.
• Perfect spline — A univariate spline of order m whose m-th derivative takes alternating values plus or minus one between successive knots.
• Polynomial differential form — An element of the commutative differential graded algebra of polynomial coordinate functions and their differentials on a standard simplex, varying simplicially with face and degeneracy maps.