Comparing Regular Knot Projections

Now that regular knot projections have been fully classified, one proceeds by "comparing" them, which means for any two such projections to decide whether they belong to equivalent or to inequivalent knots. (Two knots shall be considered equivalent if they are ambient isotopic to each other or to the mirror image to each other or to the inverse of each other or finally to the mirror image of the inverse of each other. In a previous page we discussed the reasons that forced us to broaden the notion of equivalence).

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.

1. Find the distinct regular knot projections by starting with all possible notations and eliminate ones that are undrawable or repeated.
2. On each such notation, perform all possible Reidemeister moves which do not increase the crossing number.
3. If through the previous step no notation emerges that is "preferred" to the original, store the notation into the computer's memory and assign it two numbers, one "permanent" and one "temporary". Initially they are equal to the "permanent" notation of the previous such notation plus one.
4. If exactly one more "preferrable" notation emerges, proceed to the next notation.
5. If two or more more "preferrable" notations emerge, perform enough Reidemeister moves on them until you reach notations that have been stored in the memory. Identify them as equivalent by altering their "temporary" numbers; all such notations must be assigned the minimum of their previous "temporary" numbers.
6. When you are through checking all notations, the output consists of all notations stored in the memory whose "permanent" and "temporary" numbers are the same.

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
```

Charilaos Aneziris, charilaos_aneziris@standardandpoors.com