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.
| Identifer | oai:union.ndltd.org:ndsu.edu/oai:library.ndsu.edu:10365/31814 |
| Date | January 2020 |
| Creators | Martin, Cody Michael |
| Publisher | North Dakota State University |
| Source Sets | North Dakota State University |
| Detected Language | English |
| Type | text/dissertation |
| Format | application/pdf |
| Rights | NDSU policy 190.6.2, https://www.ndsu.edu/fileadmin/policy/190.pdf |
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