A diagrammatic approach to link invariants of finite degree

Abstract

In [5] M. Polyak and O. Viro developed a graphical calculus of diagrammatic formulas for Vassiliev link invariants, and presented several explicit formulas for low degree invariants. M. Goussarov [2] proved that this arrow diagram calculus provides formulas for all Vassiliev knot invariants. The original note [5] contained no proofs, and it also contained some minor inaccuracies. This paper fills the gap in literature by presenting the material of [5] with all proofs and details, in a selfcontained form. Furthermore, a compatible coalgebra structure, related to the connected sum of knots, is introduced on the algebra of based arrow diagrams with one circle.

Tree invariants and Milnor linking numbers with indeterminacy

“…The main theorem of the current paper is a generalization of Theorem 1.1 to closed n-component links. Analogous result by Polyak and Viro, Theorem 6 in [29] (see also [27]), concerns the case n = 3.…”

“…Theorem of Goussarov [10] shows that any finite type invariant v of knots can be expressed as P v , · for a suitable choice of the arrow polynomial P v . Arrow polynomials of some low degree invariants have been computed in [27,29,30,35]. For instance, the second coefficient of the Conway polynomial c 2 (K) of a knot K, represented by a Gauss diagram G K is given by , G K (c.f.…”

On Type II Reidemeister moves of links

“…In this section, the definitions [12, Page 299, Section 2.3] of Gauss diagram and Gauss diagram formula, notations (symbols) [14, Figures 2-4] of oriented Reidemeister moves Ω * obey [12,14] except that "Gauss diagram formulas" are called "arrow diagram formulas" in [12].…”

Milnor's triple linking number and Gauss diagram formulas of 3-bouquet graphs

“…Bouquet graphs are elementary topological objects which have been well studied. However, explicit Gauss diagram formulas [2] of 3-bouquet graphs have not been very few or may be unknown.…”