Algebra of Chord Diagrams

The algebra of chord diagrams is the graded algebra obtained from the vector space over the rational numbers freely generated by chord diagrams by imposing the four-term relations (4T relations). The grading is the number of chords (Bar-Natan 1995).

Multiplication is given by cutting the oriented circles at unmarked points and joining the resulting arcs to form one oriented circle. Modulo the four-term relations, the product is independent of the chosen cuts. The diagram with no chords is the identity element for multiplication. It also carries a Hopf algebra structure.

For unframed knots, the one-term relation (framing-independence or 1T relation) also sets to zero a chord diagram having a chord that intersects no other chord. Modulo the four-term relations, it suffices to impose this relation on diagrams having a chord with consecutive endpoints in the cyclic order. Write for the degree- component of this quotient. Let be the vector space over the rational numbers of Vassiliev invariants of order at most , and set . The quotient is naturally the dual vector space of (Kneissler 1997, p. 3).

See also

Algebra of Chinese Characters, Associated Graded Space, Chord Diagram, Four-Term Relation, Graded Completion, Hopf Algebra, Jacobi Diagram, One-Term Relation, Universal Vassiliev Invariant, Vassiliev Invariant, Weight System

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References

Bar-Natan, D. "On the Vassiliev Knot Invariants." Topology 34, 423-472, 1995. https://doi.org/10.1016/0040-9383(95)93237-2.Kneissler, J. A. "The Number of Primitive Vassiliev Invariants up to Degree Twelve." 18 Jun 1997. https://arxiv.org/abs/q-alg/9706022.

Cite this as:

Weisstein, Eric W. "Algebra of Chord Diagrams." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AlgebraofChordDiagrams.html

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