Timeline for Diagrammatic proof of unique prime decomposition of knots
Current License: CC BY-SA 3.0
6 events
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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 |
added 219 characters in body
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Apr 24, 2013 at 18:21 | history | answered | Dave Futer | CC BY-SA 3.0 |