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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 σ4–1 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.