A complete algorithmic solution to this problem exists, which has unknown complexity.As first sketched using the theory of Haken manifolds by . For a more recent survey, see In practice, knots are often distinguished using a knot invariant, a "quantity" which is the same when computed from different descriptions of a knot. Important invariants include knot polynomials, knot groups, and hyperbolic invariants.
a cOmplete ALgoRItHmiC sOlutIoN to thiS pRobleM exisTs, whiCh hAs unKNowN ComPlEXITy.As FirsT SkeTCHed usINg The Theory OF HAkeN maNifolds by . foR A morE recEnT surVey, seE in PRactice, Knots are OFTen disTInGUished uSing a knoT inVarIAnt, a "QuANtIty" wHich Is ThE saMe when Computed from difFerent dEscRIpTiOns OF a kNot. ImportaNt iNVarIAnts IncLUde knot poLyNomiaLS, kNot grOupS, AnD hyperbOlIC INvarIants.