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I believe it's not known whether or not the uniform electrostatic charge potential function (as studied by Bryson, Freedman, He and Wang: http://front.math.ucdavis.edu/9301.5212) has any critical points other than the global minimum, on the space of unknots in $S^3$.

If the above were true and there were no critical points, that should in principle give you an algorithm to simplify any trivializable knot. Perhaps it could be implemented diagramatically, via a combination of things like simplifying Reidemeister moves and inversions on circles in the diagrams corresponding to conformal transformations of $S^3$.

Ivan Dynnikov has done some work on this and has a paper claiming diagrammatic "monotonic" simplification of unknot diagrams: http://arxiv.org/abs/math.GT/0208153 I went to a couple of his talks on this topic but didn't understand the core of the argument.

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I believe it's not known whether or not the uniform electrostatic charge potential function (as studied by Bryson, Freedman, He and Wang: http://front.math.ucdavis.edu/9301.5212) has any critical points other than the global minimum, on the space of unknots in $S^3$.

If the above were true and there were no critical points, that should in principle give you an algorithm to simplify any trivializable knot. Perhaps it could be implemented diagramatically, via a combination of simplifying Reidemeister moves and inversions on circles in the diagrams corresponding to conformal transformations of $S^3$.

Ivan Dynnikov has done some work on this and has a paper claiming diagrammatic "monotonic" simplification of unknot diagrams: http://arxiv.org/abs/math.GT/0208153 I went to a couple of his talks on this topic but didn't understand the core of the argument.

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I believe it's not known whether or not the uniform electrostatic charge potential function (as studied by Bryson, Freedman, He and Wang: http://front.math.ucdavis.edu/9301.5212) has any critical points other than the global minimum, on the space of unknots in $S^3$.

If the above were true and there were no critical points, that should in principle give you an algorithm to simplify any trivializable knot. Perhaps it could be implemented diagramatically, via a combination of simplifying Reidemeister moves and inversions on circles in the diagrams corresponding to conformal transformations of $S^3$.