Updated 10th September, 2011
Here we present a list of distinct welded knots up to 6 crossings produced a by computer search based on labelled peer codes, a description of which is available here. The programme that produced these tables is available as a source distribution here: vlist-19-8-10.tar
We have determined the following distinct welded knots. The orientation arrow in the following diagrams serves to indicate the edge labelled with zero for the given labelled immersion code.
w3.1 Labelled peer code: [-3 -7 -9 -1 -5]/* + - + *
Braid word: s1t2s3-s2-s2-s1t2-s3s2
w3.2 Labelled peer code: [-3 7 -9 -11 1 -5]/* * * + - +
Braid word: t1-s2t1-s1-s1t2
w4.1 Labelled peer code: [-3 -11 -1 -9 -5 -7]/* - + * - +
Braid word: s1t1-s1s2s1t1-s1-s2
w4.2 Labelled peer code: [-3 7 -11 -9 1 -5]/* + + * + +
Braid word: -s1-s2s3t2s1-s4s3t2s3s4-s3-s2
w4.3 Labelled peer code: [-3 7 -11 -9 1 -5]/* + - * + -
Braid word: -s1s2s3t2s1-s4s3t2s3s4-s3s2
w4.4 Labelled peer code: [-3 7 -11 -9 1 -5]/* + - + + *
Braid word: -s1s2s3t2s1-s4s3-s2s3s4-s3t2
w4.5 Labelled peer code: [-3 7 -9 -11 1 -5]/* + + - + *
Braid word: t1s2-s1t1s1s2
w4.6 Labelled peer code: [-5 9 -1 -11 3 -7]/+ - * + - *
Braid word: -s1-s2t3-s2s1-s4t3-s2-s3s4-s3s2
w6.1 Labelled peer code: [-3 -13 -1 -9 -11 -7 -5]/+ + + + + + *
Braid word: -s1-s2-s2-s2s1-s3-s2-s2-s2s3t2
These knots were distinguished by comparing the fixed point invariants determined by the 298 essential pairs of order less than or equal to 6 calculated from lists of finite biquandles. The fixed point invariants were calculate using the braid programme.
The search algorithm used by the programme vlist sequentially considers all immersions with n crossings and assigns labels to those immersions in all possible combinations. The resultant candidates are then checked to see if they are equivalent to an immersion with fewer crossings (by a Reidemeister I or II move) and the fixed point invariants only calculated for the remaining viable candidates.
One consequence of this approach is that having distinguished a set of welded knots with k crosings from immersions with n crossings, one cannot discount the possibility of finding another distinct knot with k crossings from an immersion having m > n crossings.
The following file shows the essential pairs (in the format used by the braid programme), the fixed point invariant calculated for each of the above knots and a summary report for the immersions with n crossings, for n=3,4,5,6.
The immersion of knot w6.1 contains seven crossings. This was distinguished from an earlier search using only the 10 essential pairs of order less than or equal to 4. There are 810852 labelled immersion codes for immersions with seven crossings and checking the full list of 298 essential pairs for these immersions would take an unreasonable length of time. However, knot w6.1 is able to be distinguished from the rest using the 10 essential pairs of the earlier search. The invariants for w6.1 are also included in the above file.
In many cases two labelled immersions cannot be distinguished by any of the 298 fixed point invariants we calculate, and it is clear that being indistinguishable (by these invariants) defines an equivalence relation on labelled immersions. It should be noted that the fixed point invariant is weak in its ability to distinguish welded knots in the sense that the size of the indistinguishable equivalence classes can be quite large.
The list given above should therefore be viewed as a list of representatives of the indistinguishable equivalence classes, rather than a definitive list of distinct welded knots. For example, the above list includes non-prime immersions: if the search is carried out to consider only prime immersions it will produce a different set of representative labelled immersions.
[1] A. Bartholomew and R. Fenn. Biquandles and Welded Knot Invariants of Small Size (to appear).