are not known to be efficiently computable for this purpose. It is not known whether the Jones polynomial or finite type invariants can detect the unknot.

Examples

It can be difficult to find a way to untangle string even though the fact it started out untangled proves the task is possible. Thistlethwaite and Ochiai provided many examples of diagrams of unknots that have no obvious way to simplify them, requiring one to temporarily increase the diagram's crossing number.

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