Fiedler type combinatorial formulas for generalized Fiedler type invariants of knots in \(M^2\times R^1\)
By using Gauss diagrams for knots and links, \textit{M. Polyak} and \textit{O. Viro} developed combinatorial formulas for finite type invariants [Int. Math. Res. Not. 1994, No.~11, 445--453 (1994; Zbl 0851.57010)]. Fiedler introduced and studied Gauss diagram invariants of low degree for knots in a trivial surface bundle \(M^2\times\mathbb R\). These invariants are parametrized by non-trivial homology classes in \(H_1(M^2\times\mathbb R)\) [\textit{T. Fiedler}, Gauss diagram invariants for knots and links. Mathematics and its Applications (Dordrecht). 532. Dordrecht: Kluwer Academic Publishers (2002; Zbl 1009.57001)]. In the paper under discussion the authors construct Gauss diagram invariants of higher order indexed by ordered collections of non-trivial free loop homotopy classes of \(M^2\times\mathbb R\). The authors show how their invariants are related in low degree to Fiedler's invariants. They also give examples of knots in the torus bundle \(T^2\times\mathbb R\) which are not separated by any degree 1 and 2 invariants but can be separated by their new invariants.
scientific article; zbMATH DE number 5597509