Simplified numerical form of universal finite type invariant of Gauss words (Q2848020)

Knots can be studied from a combinatorial point of view using what is called knot diagrams and Reidemeister moves. Another approach for studying knots was introduced by \textit{V. Turaev} [Int. Math. Res. Not. 2006, No. 25, Article ID 84098, 23 p. (2006; Zbl 1118.57009)]; [Jpn. J. Math. (3) 2, No. 1, 1--39 (2007; Zbl 1162.68034)]; [Proc. Lond. Math. Soc. (3) 95, No. 2, 360--412 (2007; Zbl 1145.57018)] in which he extended the theory of virtual knots from the aspect of Gauss codes to nanowords. Gauss words are defined as the simplest version of nanowords and homotopy classes of Gauss words are equivalent to free knots with a base points as defined in \textit{V. O. Manturov} [Sb. Math. 201, No. 5, 693--733 (2010; Zbl 1210.57010)].NEWLINENEWLINEIn [Topology Appl. 158, No. 8, 1050--1072 (2011; Zbl 1268.57009)], \textit{A. Gibson} and \textit{N. Ito} defined finite type invariants of nanophrases in a similar way to \textit{M. Goussarov} et al. [Topology 39, No. 5, 1045--1068 (2000; Zbl 1006.57005)] and obtained the finite type invariant of Gauss words of degree 4.NEWLINENEWLINEIn the paper under review, the authors obtain explicitly the finite type invariants of degree four, five and six.