Layered Venn diagrams and Brunnian links

Draw the curves of a simple Venn diagram as pieces of string, with each curve going alternately over and under at its crossings, and you get a link, a set of closed loops in space. This is the diagram's alternating link. There are two such choices, mirror images of each other. For three circles the alternating link is the Borromean rings: no two rings are linked, but the three together cannot be pulled apart. A link like that, which is not an unlink but comes apart into unlinked loops when any one loop is removed, is called Brunnian.

The question here is whether some simple symmetric 11-curve Venn diagram has a Brunnian alternating link. It is open. At 7 curves the answer is yes: Rick Mabry and Laura McCormick showed that M2, one of the simple symmetric 7-curve diagrams Grünbaum found, is Brunnian ("Branko's M2 Is Brunnian", Mathematics Magazine 93, 2020, doi:10.1080/0025570X.2020.1823767), and in Claude's computations its alternating link is Brunnian too. M2 is one of the diagrams in the 56 post, which defines simple and symmetric. At 11 curves Claude searched one family, the layered diagrams that can be drawn as cylindrical wirings, found 370,888 of them, and showed that none is Brunnian.

A diagram is layered if, for every pair of curves, one of them passes over the other at every crossing between the two. Two loops arranged that way can be pulled apart by lifting the upper one away, so in a layered diagram no two curves are linked, which a Brunnian link needs. The known symmetric Brunnian examples, with 3, 5 and 7 curves, are all layered; Claude checked each one crossing by crossing. That makes layered diagrams a reasonable place to look.

In a symmetric diagram with an odd number of curves, the rotation maps the alternating link to itself, so in a layered one, whether curve c passes over curve c+d depends only on the offset d. If c is over c+d, then c+d is under c, so the offsets d and -d are tied and only half of them are free. Choosing them gives a circulant tournament: a rule saying which curve of each pair is on top, the same after every rotation. For 11 curves that is 5 choices, or 32 tournaments, and taking the mirror image of the link reverses every choice, which leaves 16 cases.

Seven curves

Among the 56 simple symmetric 7-curve diagrams exactly two are layered, M2 and the one called Victoria, and both are monotone. Monotone means every region other than the outside and the innermost one touches a region inside one fewer curve and one inside one more; by the monotone post, having no local minimum is enough. The count of two comes from testing each diagram on the complete list. That no non-monotone one is layered was also proved directly, with one SAT formula for each of the four tournaments on 7 curves (up to mirror image) and checked unsatisfiability proofs from both encoders. Claude ran these checks, and Codex did not review them.

Victoria is not Brunnian. Removing one of its curves leaves six that do not form an unlink, which the braid test below proves. Claude's link tests on all 56 diagrams found no Brunnian one other than M2, but for some of them the verdict rests on numerical SnapPy computations, so that part is a computation, not a theorem.

Eleven curves

At 11 curves we did not search every layered diagram, only those that can be drawn as a cylindrical wiring. Picture 11 wires running around a centre like the strands of a braid, with neighbouring wires swapping places at crossings, and the pattern repeated 11 times around so that a turn of 1/11 maps the picture to itself. These diagrams are all monotone, and the link is the closure of a braid, which makes it easy to test. We have not proved the converse, that every monotone symmetric 11-curve diagram can be drawn this way, so everything below is about wirings.

Claude wrote the search. It builds a wiring level by level with exact covers and joins the two halves at the middle level. Up to rotation, reflection and turning the diagram inside out, it found 370,888 layered wiring diagrams, all for the same tournament, the one in which curve c is over curve c+d for every d from 1 to 5. For the other 15 tournaments it found none. The search took about 57 CPU hours.

To rule out Brunnian, one proper sub-collection of curves that is not an unlink is enough. A theorem of Cochran, and of Grigsby and Wehrli, says that if the closure of an m-strand braid is the m-component unlink, the braid itself is trivial. So for each diagram Claude's code found a set of 3, 4 or 10 curves whose sub-braid is not trivial. The closure of that sub-braid is then not an unlink, and since the set is a proper subset of the 11 curves, the diagram is not Brunnian. Non-triviality comes from the Lawrence-Krammer matrices of the braid group, evaluated at one fixed point modulo the prime 67,108,859: if one entry of the product differs from the identity matrix, the braid is not trivial. The checkers verify that the matrices satisfy the braid relations at that point, so the argument does not depend on the formulas being right.

Codex wrote a separate checker in C++ from the description of the file format, without running or importing Claude's code. It accepted all 370,888 witnesses in 114 CPU seconds. It rejected a deliberately corrupted input, and it was itself tested on the braid relations at 36 random parameter sets. A second, simpler checker in Python, also written from the format description, agreed on every thousandth line.

So none of the 370,888 is Brunnian. Given the braid theorem, that part is computed and independently checked.

What is not established

The weak spot is the word "all". That the 370,888 are every layered wiring diagram rests on the search code and its cross-checks, including a separate SAT encoding of the same model that found two layered diagrams, both already on the list. Codex reviewed the search, recomputed the join at the middle level, and gave what it called conditional support: the reductions in the search have to be mathematically right as reviewed, the data files have to come from the reviewed code, and the braid theorem has to hold. Nobody has re-enumerated the family with separately written search code.

Layered diagrams that are not wirings, in particular any non-monotone ones, are not covered. Whether a layered symmetric 11-curve diagram has to be monotone is open; by the monotone post, that is the same as asking whether one can have a local minimum. Diagrams that are not layered are not covered either, and only the alternating link was tested.

So whether any simple symmetric 11-curve diagram has a Brunnian alternating link stays open. A SAT search over monotone 11-curve diagrams that need not be layered was running on October 19 and had found neither a Brunnian candidate nor a proof that there is none.