Unfortunately there is no practical procedure that can directly tell whether two knots are equivalent or not, despite a claim by Haken that in principle such a method exists. We are thus forced to use two distinct procedures.
The first one performs Reidemeister moves on each projection and checks whether eventually one arrives to the other projection. If that is the case, the projections have been shown to belong to equivalent knots and no further procedure is required.
The second obtains the Alexander polynomials and then performs the "color tests" on the two projections. If they possess one or more distinct such polynomials or if they yield distinct responses to one or more such color tests, the inequivalence of the corresponding knots has been established.
If the two projections cannot be reached by Reidemeister moves and also cannot be shown inequivalent through polynomials and tests, their relation remains unresolved.
In practice the steps performed are the following.
To review the notion of the "more preferrable" notation, the reader may look here. One should notice however that if two notations have different crossing numbers, the one with the smallest crossing number is ALWAYS the most preferrable one.
Naturally, the more regular projections one considers, the more "accurate" the results, in the sense that the greater the likelihood for two equivalent knots to be identified. Usually one checks all projections whose crossing number does not exceed some cutoff parameter. Were one to set this parameter equal to infinity, one would obtain perfect result (after an infinite amount of time of course), and no need would arise to obtain any knot characteristics. For a finite cutoff parameter NF one usually obtains "inaccurate" results in the sense that equivalent knots appear more than once in the output, since their equivalence through Reidemeister moves can only be obtained through the inclusion of projections whose crossing number exceeds NF. Therefore for each NF there is a number N such that if one sets cutoff equal to NF, the results are accurate up to N crossings. For NF up to 15 the value N is equal to:
NF | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 -------------------------------------------------------------------------- N | 3 3 3 3 4 5 6 6 7 8 8 9 10 10 11 12
Here you may obtain a table of the results for various input values.
Charilaos Aneziris, firstname.lastname@example.org
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