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Apr 25, 2013 at 13:53 vote accept Daniel Moskovich
Apr 25, 2013 at 2:18 comment added Daniel Moskovich Thank you for this answer! So the answer, essentially, is that no such proof exists and that any such proof seems out of reach. So, for now anyway, prime factorization of knots and links is essentially a topological rather than a combinatorial fact.
Apr 24, 2013 at 19:23 comment added Dave Futer Ryan, that's a good point. In fact, it brings up the grey and amorphous boundary of the "diagrammatic" concept. For instance, take Menasco's proof that prime decompositions of alternating knots must be visible in the diagram. That argument is diagrammatic in flavor, but cut-and-paste topology (of the kind that Daniel seems to want to rule out) is also present. So is this in or out?
Apr 24, 2013 at 19:00 comment added Ryan Budney Alternatively, Marc Lackenby has a formalism for normal surfaces that one could view as being almost diagrammatic. He customizes a decomposition of the knot complement to a diagram in a way that allows for normal surface theory. This appears in his recent paper on the additivity of crossing number problem.
Apr 24, 2013 at 18:31 history edited Dave Futer CC BY-SA 3.0
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Apr 24, 2013 at 18:21 history answered Dave Futer CC BY-SA 3.0