3-manifold spine cyclic presentations with seldom seen Whitehead graphs

TL;DR

This paper introduces new cyclic presentations that serve as spines of closed 3-manifolds, with Whitehead graphs similar to Fractional Fibonacci and McDermott's positive cyclic presentations, often resulting in hyperbolic manifolds.

Contribution

It provides novel examples of 3-manifold spines with specific Whitehead graph types, expanding understanding of their combinatorial and geometric properties.

Findings

New cyclic presentations are spines of closed 3-manifolds.

Whitehead graphs match those of Fractional Fibonacci and McDermott's types.

Many resulting manifolds are hyperbolic.

Abstract

We consider a family of cyclic presentations and show that, subject to certain conditions on the defining parameters, they are spines of closed 3-manifolds. These are new examples where the reduced Whitehead graphs are of the same type as those of the Fractional Fibonacci presentations; here the corresponding manifolds are often (but not always) hyperbolic. We also express a lens space construction in terms of a class of positive cyclic presentations that are spines of closed 3-manifolds. These presentations then furnish examples where the Whitehead graphs are of the same type as those of the positive cyclic presentations of type $\mathfrak{Z}$, as considered by McDermott.