On braids and groups $G_n^k$

TL;DR

This paper explores the relationships between free braid groups, classical braid groups, and free groups, providing explicit homomorphisms that could lead to new braid invariants and complexity measures.

Contribution

It explicitly describes homomorphisms connecting free braid groups with classical braid groups and free groups, revealing new structural insights.

Findings

Explicit homomorphisms for k=3,4 cases

Connections enabling new braid invariants

Potential for new complexity measures

Abstract

In [V.O. Manturov, Non-reidemeister knot theory and its applications in dynamical systems, geometry, and topology, arxiv:1501.05208] the first named author gave the definition of $k$-free braid groups $G_n^k$. Here we establish connections between free braid groups, classical braid groups and free groups: we describe explicitly the homomorphism from (pure) braid group to $k$-free braid groups for important cases $k=3,4$. On the other hand, we construct a homomorphism from (a subgroup of) free braid groups to free groups. The relations established would allow one to construct new invariants of braids and to define new powerful and easily calculated complexities for classical braid groups.