Motivated by problem of determining the unknotted routes for the scaffolding strand in DNA origami self-assembly, we examine the existence and knottedness of A-trails in graphs embedded on the torus. We show that any A-trail in a checkerboard-colorable torus graph is unknotted and characterizes the existence of A-trails in checkerboard-colorable torus graphs in terms of pairs of quasitrees in associated embeddings. Surface meshes are frequent targets for DNA nanostructure self-assembly, and so we study both triangular and rectangular torus grids. We show that aside from one exceptional family, a triangular torus grid contains an A-trail if and only if it has an odd number of vertices, and that such an A-trail is necessarily unknotted. On the other hand, while every rectangular torus grid contains an unknotted A-trail, we also show that any torus knot can be realized as an A-trail in some rectangular grid. Lastly, we use a gluing operation to construct infinite families of triangular and rectangular grids containing unknotted A-trails on surfaces of arbitrary genus. We also give infinite families of triangular grids containing no unknotted A-trail on surfaces of arbitrary nonzero genus.

The strong tree property and ITP (also called the super tree property) are generalizations of the tree property that characterize strong compactness and supercompactness up to inaccessibility. That is, an inaccessible cardinal κ is strongly compact if and only if the strong tree property holds at κ, and supercompact if and only if ITP holds at κ. We present several results motivated by the problem of obtaining the strong tree property and ITP at many successive cardinals simultaneously; these results focus on the successors of singular cardinals. We describe a general class of forcings that will obtain the strong tree property and ITP at the successor of a singular cardinal of any cofinality. Generalizing a result of Neeman about the tree property, we show that it is consistent for ITP to hold at ℵn for all 2 ≤ n < ω simultaneously with the strong tree property at ℵ ω+1 ; we also show that it is consistent for ITP to hold at ℵn for all 3 < n < ω and at ℵ ω+1 simultaneously. Finally, turning our attention to singular cardinals of uncountable cofinality, we show that it is consistent for the strong and super tree properties to hold at successors of singulars of multiple cofinalities simultaneously. THE STRONG AND SUPER TREE PROPERTIES AT SUCCESSORS OF SINGULAR CARDINALS 3 Preliminaries and Branch LemmasWe begin by defining the the strong tree property and ITP.Definition 2.1. Let µ be a regular cardinal and let λ ≥ µ.Note that if µ is inaccessible, every P µ (λ) list is thin.In other words, for all x, b ∩ x ∈ L x . We say that the strong tree property holds at µ if for all λ ≥ µ, every thin P µ (λ)-list has a cofinal branch.We say ITP(µ, λ) holds if every thin P µ (λ)-list has an ineffable branch. We say ITP holds at µ if ITP(µ, λ) holds for all λ ≥ µ.Note that every ineffable branch is cofinal.Fact 2.5. [5, Lemma 3.4] Let λ ′ > λ. If every thin P µ (λ ′ )-list has a cofinal branch, then so does every thin P µ (λ). ITP, like the tree property, is usually obtained at the successor of regular cardinals by means of a lifted embedding. This will produce a branch in the generic extension containing the embedding, as described in the following lemma.Lemma 2.6. Let W be a model of set theory, and let d be a thin P κ (λ) list. Suppose that in some extension W [G] there is a generic embedding j : W → M with critical point κ such that j(κ) > λ and M λ ⊆ M . Then in W [G], d has an ineffable branch.Proof. Consider j( d). Since j(κ) > λ, we have that j ′′ λ ∈ P j(κ) (j(λ)). Define c = j( d) j ′′ λ ; that is, the entry of j( d) indexed by j ′′ λ. Let b = {α < λ | j(α) ∈ c}. We claim that b is an ineffable branch. Let U be the normal measure on P κ (λ) corresponding to j. Note that in the ultrapower by U , j ′′ λ is represented by [x → x], and so c is represented by [x → d] [x →x] = [x → d x ]. On the other hand, c = j(b) ∩ j ′′ λ. Since j(b) is represented by [x → b], c = [x → b] ∩ [x → x]. We conclude that d x = b ∩ x for U -many x. Since U is a normal measure, all measure one sets are stationary. It follows that {x | d x = b ∩...

Motivated by the problem of determining unknotted routes for the scaffolding strand in DNA origami self-assembly, we examine existence and knottedness of A-trails in graphs embedded on the torus. We show that any Atrail in a checkerboard-colorable torus graph is unknotted and characterize the existence of A-trails in checkerboard-colorable torus graphs in terms of pairs of quasitrees in associated embeddings. Surface meshes are frequent targets for DNA nanostructure self-assembly, and so we study both triangular and rectangular torus grids. We show that, aside from one exceptional family, a triangular torus grid contains an A-trail if and only if it has an odd number of vertices, and that such an A-trail is necessarily unknotted. On the other hand, while every rectangular torus grid contains an unknotted A-trail, we also show that any torus knot can be realized as an A-trail in some rectangular grid. Lastly, we use a gluing operation to construct infinite families of triangular and rectangular grids containing unknotted A-trails on surfaces of arbitrary genus. We also give infinite families of triangular grids containing no unknotted A-trail on surfaces of arbitrary nonzero genus.