# Generalized Fibonacci groups H(r,n,s) that are connected Labelled Oriented Graph groups

## TL;DR

This paper classifies when generalized Fibonacci groups H(r,n,s) are connected Labelled Oriented Graph groups, linking them to knot groups and using circulant matrices to analyze their properties.

## Contribution

It provides an almost complete classification of H(r,n,s) groups as connected LOG groups, identifying all torus knot groups and the infinite cyclic group within this class.

## Key findings

- All torus knot groups are connected LOG groups.
- H(r,n,s) is a 2-generator knot group if and only if it is a torus knot group.
- Conjecture: only torus knot groups and the infinite cyclic group are connected LOG groups.

## Abstract

The class of connected LOG (Labelled Oriented Graph) groups coincides with the class of fundamental groups of complements of closed, orientable 2-manifolds embedded in S^4, and so contains all knot groups. We investigate when Campbell and Robertson's generalized Fibonacci groups H(r,n,s) are connected LOG groups. In doing so, we use the theory of circulant matrices to calculate the Betti numbers of their abelianizations. We give an almost complete classification of the groups H(r,n,s) that are connected LOG groups. All torus knot groups and the infinite cyclic group arise and we conjecture that these are the only possibilities. As a corollary we show that H(r,n,s) is a 2-generator knot group if and only if it is a torus knot group.