On the algebra of arrow diagrams (Q1587208)

The author develops further the Gauss diagram approach to finite-type link invariants. He defines an algebra \(A\) of arrow diagrams modulo 6T relation ; this algebra is a counterpart of the chord diagrams approach. Remark that a theory of finite-type invariants underlying this algebra was introduced by Goussarov, Polyak and Viro in '98. Since the 6T relation is related to the classical Yang-Baxter equation and any solution of the classical Yang-Baxter equation leads to a homomorphism from \(A\) to \(\mathbb R\), it allows to produce real-valued link invariants from any \(A\)-valued one. Next, the author introduces an algebra \(G\) of acyclic arrow graphs modulo an appropriate oriented version of the STU relation, which is isomorphic to \(A\). As an application he considers an appropriate link homotopy version \(A_k\) of \(A\), and constructs a Gauss diagram invariant of string links up to link-homotopy, with values both in \(A_k\) and \(\mathbb R\). This construction gives the universal Milnor's link-homotopy \(\mu\)-invariant.