Related papers: Geometry of Calugareanu's theorem
Geometry is wavy: even at the purely geometric level (no particular theory chosen), curvature satisfies a covariant quasilinear wave equation. In Riemannian geometry equipped with the Levi-Civita connection, the Riemann curvature tensor…
We propose localization techniques for computing Gromov-Witten invariants of maps from Riemann surfaces with boundaries into a Calabi-Yau, with the boundaries mapped to a Lagrangian submanifold. The computations can be expressed in terms of…
Milnor's triple linking numbers of a link in the 3-sphere are interpreted geometrically in terms of the pattern of intersections of the Seifert surfaces of the components of the link. This generalizes the well known formula as an algebraic…
Just how many different connected shapes result from slicing a cube along some of its edges and unfolding it into the plane? In this article we answer this question by viewing the cube both as a surface and as a graph of vertices and edges.…
The Kauffman bracket skein algebra is a quantization of the algebra of regular functions on the $SL_2$ character variety of a topological surface. We realize the skein algebra of the $4$-punctured sphere as the output of a mirror symmetry…
It is shown that there exists a twistor space on the $n$-fold connected sum of complex projective planes $n\mathbb{CP}^2$, whose algebraic dimension is one and whose general fiber of the algebraic reduction is birational to an elliptic…
We study simple, knotted and linked torus windings that are made of tubes of finite thickness. Knots which have the shortest rope length are often denoted ideal structures. Conventionally, the ideal structure are found by rope shortening…
Band engineering in twisted bilayers of the five generic two-dimensional Bravais networks is demonstrated. We first derive symmetry-based constraints on the interlayer coupling, which helps us to predict and understand the shape of the…
The convex hull of a set K in space consists of points which are, in a certain sense, "surrounded" by K. When K is a closed curve, we define its higher hulls, consisting of points which are "multiply surrounded" by the curve. Our main…
The six-dimensional (2,0) theory formulated in the \Omega-background gives rise to two-dimensional effective degrees of freedom. By compactifying the theory on the circle fibers of two cigar-like manifolds, we find that a natural candidate…
We show how the description of a shear-free ray congruence in Minkowski space as an evolving family of semi-conformal mappings can naturally be formulated on a finite graph. For this, we introduce the notion of holomorphic function on a…
Combinatorial rigidity theory seeks to describe the rigidity or flexibility of bar-joint frameworks in R^d in terms of the structure of the underlying graph G. The goal of this article is to broaden the foundations of combinatorial rigidity…
Topological invariants such as winding numbers and linking numbers appear as charges of topological solitons in diverse nonlinear physical systems described by a unit vector field defined on two and three dimensional manifolds. While the…
Symmetry of geometrical figures is reflected in regularities of their algebraic invariants. Algebraic regularities are often preserved when the geometrical figure is topologically deformed. The most natural, intuitively simple but…
This paper constructs a graded-commutative, associative, differential Transverse Intersection Algebra TIA {on the torus (in any dimension) with its cubical decomposition by using a probabilistic wiggling interpretation. This structure…
A natural problem in combinatorial rigidity theory concerns the determination of the rigidity or flexibility of bar-joint frameworks in $\mathbb{R}^d$ that admit some non-trivial symmetry. When $d=2$ there is a large literature on this…
A curve $\gamma$ in a Riemannian manifold $M$ is three-dimensional if its torsion (signed second curvature function) is well-defined and all higher-order curvatures vanish identically. In particular, when $\gamma$ lies on an oriented…
We show that the intrinsic angular momentum of matter in curved spacetime requires the metric-affine formulation of gravity, in which the antisymmetric part of the affine connection (the torsion tensor) is not constrained to be zero but is…
In this paper we show that the number of all 1/2-BPS branes in string theory compactified on a torus can be derived by universal wrapping rules whose formulation we present. These rules even apply to branes in less than ten dimensions whose…
We give explicit formulae for fringe lengths of the Calegari-Walker Ziggurats -- i.e. graphs of extremal rotation numbers associated to positive words in free groups. These formulae reveal (partial) integral projective self-similarity in…