The Kakimizu Complex for a knot is what you get by taking the definition of the curve complex for a surface and replacing loops in the surface with minimal genus Seifert surfaces for . It consists of a vertex for each isotopy class of minimal genus Seifert surface for with edges connecting any two vertices with disjoint representatives, and simplices spanning larger collections of pairwise disjoint surfaces. This complex turns out to be contractible [1] and for atoroidal knots, it’s finite [2]. But something I found quite surprising is an example by Jessica Banks [3] whose Kakimizu complex is locally infinite. This knot is relatively simple and you can see a picture of it below the fold.

The knot is a satellite over the trefoil with the property that inside the solid torus defined by the trefoil, is homologically trivial. (Correction: In Bank’s minimal example, there is one fewer full twists in the band between the parallel trefoils.) That means that it has a Seifert surface contained in the solid torus (shown on the right). But it also has a minimal genus Seifert surface that cuts through the solid torus (shown on the left). In the picture, the colors indicate the two “sides” of the Seifert surface.

If you take the correct Haken sum of the surface on the left with a copy of the incompressible torus, you get a new minimal genus Seifert surface ~~that is disjoint from the two surfaces shown here~~. If you Haken sum the resulting surface with a second copy of the incompressible torus, the new surface will intersect the original surface on the left non-trivially, but will be disjoint from the surface on the right.

If you continue taking Haken sums, all the resulting surfaces will be disjoint from the original surface shown on the right. This follows from the fact that the right surface is disjoint from the incompressible torus. Banks shows that the Haken summed surfaces are isotopically distinct, so in the Kakimizu complex the vertex corresponding to the surface on the right has infinitely many edges coming out of it. ~~The portion of the Kakimizu complex spanned by these surfaces is isomorphic to a cone over the line with edges between consecutive integers.~~ (Correction: After a brief e-mail discussion with Jessica Banks, I think this statement is incorrect. It seems that the complex for this knot is a cone over infinitely many discrete points, but I’m still worried I’m missing something.)

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Maybe it’s not so surprising? Knots whose complements have non-trivial JSJ decompositions, their symmetry groups (\pi_0 Diff(S^3 preserving the knot) ) tends to be infinite. These groups act on the Kakimizu Complex, so that’s a reason for this complex to be big.

Comment by Ryan Budney — June 15, 2011 @ 12:41 pm |

Right, it’s not surprising that the Kakimizu complex is infinite. But it was an open problem whether it was locally infinite, i.e. whether every vertex touched a finite number of edges. This turns out to be relatively rare, and I think Banks gives a characterization of the knots whose complexes are locally infinite.

Comment by Jesse Johnson — June 15, 2011 @ 12:53 pm |

One way to demonstrate local non-finiteness then would be to find an infinite-order symmetry of the knot complement that fixes one Seifert surface. I would guess generically such a symmetry would have to be infinite-order on any “adjacent” surface. I haven’t had time to read the Banks paper. The Whitehead double of a trefoil does have an infinite symmetry group. Is that what happens? I’m not seeing a Seifert surface fixed by a symmetry, though.

Comment by Ryan Budney — June 15, 2011 @ 1:14 pm |

In this case, there is a symmetry that corresponds to spinning or twisting around the incompressible torus. Think of a Dehn twist cross a circle. This will fix the Seifert surface inside the solid torus, but there also has to be a minimal genus Seifert surface that is not fixed by this automorphism. In this case, it takes the other Seifert surface to its Haken sum with two copies of the torus.

Comment by Jesse Johnson — June 15, 2011 @ 2:04 pm |

Nice diagrams, by-the-way!

Comment by Ryan Budney — June 17, 2011 @ 6:22 pm |

[…] Jesse Johnson: Spinning Around the Kakimizu Complex […]

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