Further Affinities with Lie algebra

(the short list)

Lie Theory Chord Diagrams / Knots
U(g) knots
U(g)m links
the product m of U(g) connected sums:
the coproduct Δ of U(g) doubling:
the bracket [ , ] of g the Goussarov-Habiro theory of claspers
the Poincaré-Birkhoff-Witt theorem formal PBW, eigenspaces of cabling
the Harish-Chandra-Duflo isomorphism Wheeling, 1+1=2:
Chern-Simons theory configuration space integrals
integration on g formal Gaussian integration, 3-manifolds
Lie bialgebras directed diagrams, virtual knots
associators and the pentagon, 6j-symbols paranthesized tangles, knotted trivalent graphs