If the notation is to be useful AT ALL, it has to be consistent regardless of starting location. Or it has to specify how to identify the starting location and that’s beyond my understanding of knot projections.
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Not a knot theorist as such but: As Adams says in 2.2, Dowker describes *a* *projection* of a knot. Your transformations are different projections. He goes on to say pick any crossingand label it 1. Different starting points can give different sequences for the same projection.
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That jives with my experience- I but then what possible use is it? I get there is not yet a perfect invariant, but if this honestly only maps a projection to that same projection, then why bother? Certainly won’t help me ID a prime knot anymore than an image.
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Every knot diagram has several different Dowker notations, and each Dowker notation only describes the diagram up to a combination of (1) a mirror image, and deformations (isotopies) * on the sphere S² *
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