Knot Groups and Bi-Orderable HNN Extensions of Free Groups

Abstract

Suppose K is a fibered knot with bi-orderable knot group. We perform a topological winding operation to half-twist bands in a free incompressible Seifert surface Σ of K. This results in a Seifert surface Σ' with boundary that is a non-fibered knot K'. We call K a fibered base of K'. A fibered base exists for a large class of non-fibered knots. We prove K' has a bi-orderable knot group if Σ' is obtained from applying the winding operation to only one half-twist band of Σ. Utilizing a Seifert surface gluing technique, we obtain HNN extension group presentations for both knot groups that differ by only one relation. To show the knot group of K' is bi-orderable, we apply the following: Let G be a bi-ordered free group with order preserving automorphism ɑ. It is well known that the semidirect product ℤ ×ɑG is a bi-orderable group. If X is a basis of G, a presentation of ℤ ×ɑG is ⟨ t,X | R ⟩, where the relations are R = {txt-1}ɑ(x)-1 : x ∈ X}. If R' is any subset of R, we prove that the group H =⟨ t,X | R' ⟩ is bi-orderable. H is a special case of an HNN extension of G. Finally, we add new relations to the group presentation of H such that bi-orderability is preserved.