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The surface code is designed to suppress errors in quantum computing hardware and currently offers the most believable pathway to large-scale quantum computation. The surface code requires a 2-D array of nearest-neighbor coupled qubits that…
This paper proves an elementary topological fact about closed curves on surfaces, namely that by carefully smoothing an intersection point, one can reduce self-intersection by exactly $1$. This immediately implies a positive answer to a…
In this paper, we want to study the link between the presence of compact objects with some analytic structure and the global geometry of a weakly complete surface. We begin with a brief survey of some now classic results on the local…
This paper proposes a simple technique of curve and surface construction with B-splines. Given a control polygon or a control mesh together with node ordinates corresponding to all control points, a rational curve or surface is obtained by…
Understanding body part geometry is crucial for precise medical diagnostics. Curves effectively describe anatomical structures and are widely used in medical imaging applications related to cardiovascular, respiratory, and skeletal…
We investigate complete non-orientable minimal surfaces of finite total curvature in $\mathbb{R}^3$ such that their ends are foliated by closed lines of curvature. This condition on the ends is necessary if they have a piece inside some…
In this work we study homeomorphisms of closed orientable surfaces homotopic to the identity, focusing on the existence of non-contractible periodic orbits. We show that, if $g$ is such a homeomorphism, and if $\hat g$ is its lift to the…
This paper solves the problem of computing conformal structures of general 2-manifolds represented as triangle meshes. We compute conformal structures in the following way: first compute homology bases from simplicial complex structures,…
A well-known and difficult problem in computational number theory and algebraic geometry is to write down equations for branched covers of algebraic curves with specified monodromy type. In this article, we present a technique for computing…
We study the problem of path planning with soft homology constraints on a surface topologically equivalent to a disk with punctures. Specifically, we propose an algorithm, named $\Hstar$, for the efficient computation of a path homologous…
We prove that in the non-orientable setting, the minimal stretch factor of a pseudo-Anosov homeomorphism of a surface of genus $g$ with a fixed number of punctures is asymptotically on the order of $\frac{1}{g}$. Our result adapts the work…
A viable and still unproved conjecture states that, if $X$ is a smooth algebraic surface and $C$ is a smooth algebraic curve in $X$, then $C$ realizes the smallest possible genus amongst all smoothly embedded $2$-manifolds in its homology…
The Epstein-Baer theory of curve isotopies is basic to the remarkable theorem that homotopic homeomorphisms of surfaces are isotopic. The groundbreaking work of R. Baer was carried out on closed, orientable surfaces and extended by D. B. A.…
We present an algorithm for computing curves and families of curves of prescribed degree and geometric genus on real rational surfaces.
We provide a theoretical study of Algebraic Geometry codes constructed from abelian surfaces defined over finite fields. We give a general bound on their minimum distance and we investigate how this estimation can be sharpened under the…
We consider unbounded curves without endpoints. Isomorphism is equivalence up to translation. Self-avoiding plane-filling curves cannot be periodic, but they can satisfy the local isomorphism property: We obtain a set $\Omega $ of coverings…
The notion of graph covers is a discretization of covering spaces introduced and deeply studied in topology. In discrete mathematics and theoretical computer science, they have attained a lot of attention from both the structural and…
Consider a graph drawn on a surface (for example, the plane minus a finite set of obstacle points), possibly with crossings. We provide an algorithm to decide whether such a drawing can be untangled, namely, if one can slide the vertices…
We define a cone curve to be a reduced sextic space curve which lies on a quadric cone and does not go through the vertex. We classify families of bitangent planes of cone curves. The methods we apply can be used for any space curve with…
We investigate geometric properties of surfaces given by certain formulae. In particular, we calculate the singular curvature and the limiting normal curvature of such surfaces along the set of singular points consisting of singular points…