# DNA Origami and Unknotted A-trails in Torus Graphs

## TL;DR

This paper investigates the existence and knottedness of A-trails in graphs embedded on tori, with implications for DNA origami self-assembly, revealing conditions for unknotted trails in various torus grid types.

## Contribution

It characterizes when A-trails are unknotted in torus graphs, especially in triangular and rectangular grids, and constructs infinite families with specific trail properties.

## Key findings

- Checkerboard-colorable torus graphs have unknotted A-trails.
- Triangular torus grids have A-trails iff they have an odd number of vertices.
- Rectangular torus grids always contain unknotted A-trails, and can realize any torus knot.

## Abstract

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 A-trail 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.