Narrow Results

We study the 4-bialgebra of graphs and the bialgebra of 4-invariants introduced by S. K. Lando. Our main goal is the investigation of the relationship between 4-invariants of graphs and weight systems arising in the theory of finite order invariants of knots. In particular, we show that the corank of the adjacency matrix of a graph leads to the weight system coming from the defining representation of the Lie algebra gl(N).

The number |K| of non-isotopic framed knots that correspond to a given unframed knot K ⊂ S³ is infinite. This follows from the existence of the self-linking number slk of a zero homologous framed knot. We use the approach of Vassiliev–Goussarov invariants to construct "affine self-linking numbers" that are extensions of slk to the case of nonzero homologous framed knots in 3-manifolds.

As a corollary we get that |K| = ∞ for all knots in an oriented (not necessarily compact) 3-manifold M that is not realizable as a connected sum (S¹ × S²)# M′. This result for compact manifolds was first stated by Hoste and Przytycki. They referred to the works of McCullough for the idea of the proof, however to the best of our knowledge prior to this work the proof of this fundamental fact was not given in literature or in a preprint form. Our proof is based on different ideas. For M = (S¹ × S²)# M′ we construct K in M such that |K| = 2 ≠ ∞.