Narrow Results

In this paper we study the ℤ-module A₂ of two-chord diagrams for knots with zero winding number in the solid torus KST₀, which is needed in studying the type-two invariants for knots in KST₀. We show that this module (or abelian group), which is given as a presentation with infinite number of generators and an infinite number of relations, is a free infinitely generated module. Moreover, we show that this module is isomorphic to the direct sum of three free modules that are easier to understand.

We introduce a natural filtration in the space of knots and singular knots in the solid torus, and start the study of the type-two Vassiliev invariants with respect to this filtration. The main result of the work states that any such invariant within the second term of this filtration in the space of knots with zero winding number is a linear combination of seven explicitly described Gauss diagram invariants. This introduces a basis (and a universal invariant) for the type-two Vassiliev invariants for knots with zero winding number. Then we formalize the problem of exploring the set of all type-two invariants for knots with zero winding number.