3-braids with 4-valent vertices and knot invariant

Abstract

We construct a picture-valued invariant of classical knots by using the group $G_n^3$ and the plat closure of braids. We introduce a modification of the group $G_n^3$ and define a map from the braid group to framed 6-valent graphs with leaves. Each 6-valent vertex corresponds to the moment when three points become collinear. By using this construction, we obtain a knot invariant valued in equivalence classes of graphs modulo local moves. Our invariant provides a new graphical approach to the study of classical knots.