We investigate fat Hoffman graphs with smallest eigenvalue at least −3, using their special graphs. We show that the special graph S(H) of an indecomposable fat Hoffman graph H is represented by the standard lattice or an irreducible root lattice. Moreover, we show that if the special graph admits an integral representation, that is, the lattice spanned by it is not an exceptional root lattice, then the special graph S − (H) is isomorphic to one of the Dynkin graphs A n , D n , or extended Dynkin graphsà n orD n .

We study knots of order 2 in the grope filtration {G h } and the solvable filtration {F h } of the knot concordance group. We show that, for any integer n ≥ 4, there are knots generating a Z ∞ 2 subgroup of G n /G n.5 . Considering the solvable filtration, our knots generate a Z ∞ 2 subgroup of F n /F n.5 (n ≥ 2) distinct from the subgroup generated by the previously known 2-torsion knots of Cochran, Harvey, and Leidy. We also present a result on the 2-torsion part in the Cochran, Harvey, and Leidy's primary decomposition of the solvable filtration.

Abstract. We give an example of a 3-component smoothly slice boundary link, each of whose components has a genus one Seifert surface, such that any metaboliser of the boundary link Seifert form is represented by 3 curves on the Seifert surfaces that form a link with nonvanishing Milnor triple linking number. We also give a generalisation to m-component links and higher Milnor invariants. We prove that our examples are ribbon and that all ribbon links are boundary slice.

In this paper, we show that all fat Hoffman graphs with smallest eigenvalue at least −1−τ , where τ is the golden ratio, can be described by a finite set of fat (−1 − τ )-irreducible Hoffman graphs. In the terminology of Woo and Neumaier, we mean that every fat Hoffman graph with smallest eigenvalue at least −1−τ is an H-line graph, where H is the set of isomorphism classes of maximal fat (−1−τ )-irreducible Hoffman graphs. It turns out that there are 37 fat (−1−τ )-irreducible Hoffman graphs, up to isomorphism.