The group $G_{n}^{2}$ and Invariants of Free Knots Valued in Free Groups

TL;DR

This paper introduces a new invariant for free links and tangles valued in free products of  groups, extending previous work to handle more complex link structures with crossings and providing a method to classify free links.

Contribution

It constructs an invariant for free links and tangles using group elements associated with crossings, generalizing previous invariants to more complex link configurations.

Findings

Defined an invariant of free links valued in free products of  groups

Extended the approach to handle free tangles with complex crossings

Provided a method to classify free links using group-based invariants

Abstract

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.