  In the present paper, we define an invariant of free links valued in a free
product of some copies of $\mathbb{Z}_{2}$. In \cite{Ma2} the second named
author constructed a connection between classical braid group and group
presentation generated by elements corresponding to horizontal trisecants. This
approach does not apply to links nor tangles because it requires that when
counting trisecants, we have the same number of points at each level. For
general tangles, trisecants passing through one component twice may occur. Free
links can be obtained from tangles by attaching two end points of each
component. We shall construct an invariant of free links and free tangles
valued in groups as follows: we associate elements in the groups with 4-valent
vertices of free tangles(or free links). For a free link with enumerated
component, we `read' all the intersections when traversing a given component
and write them as a group element. The problem of `pure crossings' of a
component with itself by using the following statement: {\em if two diagrams
with no pure crossings are equivalent then they are equivalent by a sequence of
moves where no intermediate diagram has a pure crossing.} This statement is a
result of a sort that an equivalence relation within a subset coincides with
the equivalence relation induced from a larger set and it is interesting by
itself.
