Related papers: DNA Origami and Unknotted A-trails in Torus Graphs
This monograph derives direct and concrete relations between colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link complements. Under mild diagrammatic hypotheses that arise naturally in the study…
The celebrated Erd\H{o}s-P\'{o}sa Theorem, in one formulation, asserts that for every $c\geq 1$, graphs with no subgraph (or equivalently, minor) isomorphic to the disjoint union of $c$ cycles have bounded treewidth. What can we say about…
We prove that symplectic twist maps defined on the cotangent bundle of the d-dimensional torus that have no conjugate points are $C^0$ integrable, i.e. the cotangent bundle is continuously foliated by a family of invariant Lagrangian…
We relate invariants of torus knots to the counts of a class of lattice paths, which we call generalized Schr\"oder paths. We determine generating functions of such paths, located in a region determined by a type of a torus knot under…
A graph is rectilinear planar if it admits a planar orthogonal drawing without bends. While testing rectilinear planarity is NP-hard in general (Garg and Tamassia, 2001), it is a long-standing open problem to establish a tight upper bound…
The Gyarfas-Sumner conjecture asserts that if H is a tree then every graph with bounded clique number and very large chromatic number contains H as an induced subgraph. This is still open, although it has been proved for a few simple…
We present a modular DNA origami design approach to address the challenges of assembling geometrically complex nanoscale structures, including those with nonuniform Gaussian curvature. This approach features a core structure that completely…
The algebraic genus of a knot is an invariant that arises when one considers upper bounds for the topological slice genus coming from Freedman's theorem that Alexander polynomial one knots are topologically slice. This paper develops…
Numerical simulations indicate that there exist conformations of the unknot, tied on a finite piece of rope, entangled in such a manner, that they cannot be disentangled to the torus conformation without cutting the rope. The simplest…
In 2003, Ozsv\'ath, Szab\'o, and Rasmussen introduced the $\tau$ invariant for knots, and in 2011, Sarkar published a computational shortcut for the $\tau$ invariant of knots that can be represented by diagonal grid diagrams. Previously,…
Cai et al.\ proved that a toroidal graph $G$ without $6$-cycles is $5$-choosable, and proposed the conjecture that $\textsf{ch}(G) = 5$ if and only if $G$ contains a $K_{5}$ [J. Graph Theory 65 (2010) 1--15], where $\textsf{ch}(G)$ is the…
We describe a construction of invariant train tracks with irreducible transition matrix for pseudo-Anosov homeomorphisms. This fills what seems to be a gap in the literature concerning the existence of such train tracks. The construction…
The weak variant of Hanani-Tutte theorem says that a graph is planar, if it can be drawn in the plane so that every pair of edges cross an even number of times. Moreover, we can turn such a drawing into an embedding without changing the…
A genus-1 tangle G is an arc properly embedded in a standardly embedded solid torus S in the 3-sphere. We say that a genus-1 tangle embeds in a knot K in S^3 if the tangle can be completed by adding an arc exterior to the solid torus to…
Estimating the structure of directed acyclic graphs (DAGs, also known as Bayesian networks) is a challenging problem since the search space of DAGs is combinatorial and scales superexponentially with the number of nodes. Existing approaches…
In this article we present theoretical and computational results on the existence of polyhedral embeddings of graphs. The emphasis is on cubic graphs. We also describe an efficient algorithm to compute all polyhedral embeddings of a given…
We numerically examine the self-dual solutions of self-intersecting strings immersed in four dimensions. We find that open torus knots have topologies that can support monopole/anti-monopole as well as q-qbar production and annihilation. We…
A rooted acyclic digraph N with labelled leaves displays a tree T when there exists a way to select a unique parent of each hybrid vertex resulting in the tree T. Let Tr(N) denote the set of all trees displayed by the network N. In general,…
The dusty torus plays a vital role in unifying active galactic nuclei (AGNs). However, the physical structure of the torus remains largely unclear. Here we present a systematical investigation of the torus mid-infrared (MIR) spectroscopic…
Two-dimensional (2D) moir\'e superlattices have emerged as a versatile platform for uncovering exotic quantum phases, many of which arise in bilayer systems exhibiting Archimedean tessellation patterns such as triangular, hexagonal, and…