Method braid of class Link loops when the Link contains loops

[soehms]

The following link L is ambient isotopic to the treefoil knot but has one additional loop.

sage: L = Link([[1, 7, 2, 6], [3, 1, 4, 8], [5, 5, 6, 4], [7, 3, 8, 2]])
sage: L.regions()`
[[-5], [8, 4, 6, 2], [7, -2], [3, -8], [1, -6, 5, -4], [-1, -3, -7]]
sage: L.seifert_circles()
[[5], [1, 7, 3], [2, 8, 4, 6]]

Invoking L.braid() does not terminate. The problem already occurs on the first application of a Vogel move which leads to an extended pd_code = [[10, 7, 2, 6], [3, 1, 4, 8], [11, 5, 6, 4], [7, 3, 8, 2], [12, 9, 5, 1], [11, 9, 12, 10]]. A diagram of the corresponding link looks like this:

The labels C1, C2 = [3, 1, 4, 8], D = [11, 9, 12, 10] and E = [12, 9, 5, 1] correspond to the variables in the code and the numbers to the edges. In case of C1the correct list of edges would be [5, 11, 6, 4] but the code produces [11, 5, 6, 4]. This is caused by usage of the index method of the list class which always returns the first occurrence of a value in the list, i.e. by the line:

C2[C2.index(b)] = newedge + 2

To fix it I replace the method by a local function:

        def idx(cross, edge):
            r"""
            Return the index of an edge in a crossing taking loops into account.
            A loop appears as an edge which occurs twice in the crossing.
            In all cases the second occurrence is the correct one needed in
            the Vogel algorithm.
            """
            i = cross.index(edge)
            if cross.count(edge) > 1:
                return cross.index(edge, i+1)
            else:
                return i

In all cases the second occurrence is the correct one needed in the Vogel algorithm.

The according argumentation goes as follows:

If the loop is clockwise oriented the edge has a positive sign in the region, else a negative sign. In the first case the outgoing side of the edge always occurs after the incoming one in the PD-code (edge 1 in the first picture below and edge 2 in the second). In the second case the incoming side of the edge always occurs after the outgoing one in the PD-code (edge 2 in the first picture below and edge 1 in the second).

Thus, this corresponds to the fact that in the first case a and b leave the crossings C1 and C2 (in the original link) whereas they arrive there in the second case.

Another way to fix the issue would be to remove all loops from the diagram before starting the Vogel algorithm. But this would harm regular isotopy which might violate the expectation of the user. Thus, I implement this as an optional feature. This gives the user the possibility to obtain a braid of smaller index if he is fine with ambient isotopy.

📝 Checklist

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• I have linked a relevant issue or discussion.
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⌛ Dependencies

[soehms]

None of the failures from the bot-logfile appear to be related to this PR.

[tscrim]

Thank you.

Just one small additional change; once done, you can set a positive review.

[soehms]

Thank you.

Just one small additional change; once done, you can set a positive review.

Thanks!

[github-actions]

Documentation preview for this PR (built with commit f68c72a; changes) is ready! 🎉