Mathematics, Department of


First Advisor

Mark Brittenham

Second Advisor

Susan Hermiller

Date of this Version



A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy, Major: Mathematics, Under the Supervision of Professors Mark Brittenham and Susan Hermiller. Lincoln, Nebraska: August, 2019

Copyright 2019 Katie Tucker


An oriented n-component link is a smooth embedding of n oriented copies of S1 into S3. A diagram of an oriented link is a projection of a link onto R2 such that there are no triple intersections, with notation at double intersections to indicate under and over strands and arrows on strands to indicate orientation. A local move on an oriented link is a regional change of a diagram where one tangle is replaced with another in a way that preserves orientation. We investigate the local moves t3 and t4, which are conjectured to be an unlinking set (i.e., turns the link into an unlink) on oriented links (Kirby Problem List # 1.59(4)). Using combinatorial and computational methods, we show all oriented links with braid index at most 5, except for possibly the link formed from the closure of (σ1 σ2–1 σ3 σ41 σ3 σ2–1)3, are t3, t4 unlinkable. We extend these methods to oriented links with braid index 6 and crossing number at most 12.

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