A table of genus two handlebody-knots with seven crossings
We enumerate all genus two handlebody-knots with seven crossings, up to mirror image, extending the Ishii-Kishimoto-Moriuchi-Suzuki table.
math.GT↗
arXiv subjects
Giovanni Bellettini
Publications and source records attributed to Giovanni Bellettini.
Handlebody-knots obtained by tangle replacement
We study tangle replacement in the context of handlebody-knots. The main results classify the handlebody-knots obtained by performing tangle replacement on the prime spatial handcuff graph with four crossings, and determine their symmetry. The work is motivated by the study of small crossing handlebody-knots that are difficult to distinguish with computational invariants and by their chirality problem.
math.GT↗
The Role of Input Dimensionality in the Emergence and Targeted Control of Adversarial Examples
Several theoretical works have tried to explain the adversarial vulnerability of deep neural networks through properties of high-dimensional geometry. However, the assumptions underlying these works are rarely examined empirically, and systematic evidence remains limited. In this work, we present a systematic study of the role of input dimensionality in both the emergence and the targeted control of adversarial examples. We first analyse the scope and limitations of existing theoretical frameworks based on concentration of measure, showing that real image classes exhibit strong empirical localization, beyond what such theories typically assume. We then conduct an extensive empirical evaluation across hierarchical image datasets spanning a wide range of input dimensionalities and diverse neural architectures. Our results consistently show that adversarial examples become easier to construct as dimensionality increases. We also investigate how input dimensionality affects the additional difficulty of crafting targeted adversarial examples. In particular, we provide theoretical arguments showing that high-dimensional geometry implies that enforcing a specific target label entails only a limited additional distortion compared to untargeted attacks. We corroborate this insight through extensive experiments, demonstrating that the gap between targeted and untargeted perturbations remains small and further narrows as input dimensionality increases. While, taken together, our findings establish high input dimensionality as a fundamental factor underlying the emergence and targeted control of adversarial examples, whether this phenomenon primarily arises from the interplay between high-dimensional geometry and data distributions or from the architectural properties of deep neural networks remains an open question.
stat.ML↗
Concentration effects and $Γ$-limit for the elastica functional for open and closed curves
We study the $Γ$-convergence of a class of elastica-type energies defined on immersed planar curves and depending on a small positive parameter $ε$. As $ε\to 0^+$, sequences with equibounded energy develop concentration phenomena in the curvature, leading to the emergence of singularities described by atomic measures. This naturally gives rise to a limiting framework in terms of pointed curves, consisting of a curve together with a measure encoding curvature concentration. We characterize the first-order $Γ$-limit in two settings: for immersed open curves with fixed endpoints and boundary conditions on the tangents, and for immersed closed curves of prescribed length. In both cases, the limiting energy depends only on the number of concentration points and is expressed as a sum of contributions, each given by an integer multiple of $2π$. A key feature of the problem is that the rescaled energies exhibit a structure closely related to one-dimensional Modica--Mortola type functionals.
math.AP↗
$Γ$-convergence of free discontinuity problems for circle-valued maps in the linear regime
We investigate the $Γ$-convergence of Ambrosio-Tortorelli type-functionals for circle valued functions, in the case of volume terms with linear growth. We show the emergence of a non-local $Γ$-limit, which is due to the topological structure of the target space, and discuss compactness of minimal liftings. Our results extend the analysis of a previous work on the quadratic case.
math.AP↗
A fourth-order regularization of the curvature flow of immersed plane curves with Dirichlet boundary conditions
We consider a fourth-order regularization of the curvature flow for an immersed plane curve with fixed boundary, using an elastica-type functional depending on a small positive parameter $\varepsilon$. We show that the approximating flow smoothly converges, as $\varepsilon \to 0^+$, to the curvature flow of the curve with Dirichlet boundary conditions for all times before the first singularity of the limit flow.
math.AP↗
Crystalline elastic flow of polygonal curves: long time behaviour and convergence to stationary solutions
Given a planar crystalline anisotropy, we study the crystalline elastic flow of immersed polygonal curves, possibly also unbounded. Assuming that the segments evolve by parallel translation (as it happens in the standard crystalline curvature flow), we prove that a unique regular flow exists until a maximal time when some segments having zero crystalline curvature disappear. Furthermore, for closed polygonal curves, we analyze the behaviour at the maximal time, and show that it is possible to restart the flow finitely many times, yielding a globally in time evolution, that preserves the index of the curve. Next, we investigate the long-time properties of the flow using a Lojasiewicz-Simon-type inequality, and show that, as time tends to infinity, the flow fully converges to a stationary curve. We also provide a complete classification of the stationary solutions and a partial classification of the translating solutions in the case of the square anisotropy.
math.AP↗
A De Giorgi conjecture on the regularity of minimizers of Cartesian area in 1D
We prove a $C^{1,1}$-regularity of minimizers of the functional $$ \int_I \sqrt{1+|Du|^2} + \int_I |u-g|ds,\quad u\in BV(I), $$ provided $I\subset\mathbb{R}$ is a bounded open interval and $\|g\|_\infty$ is sufficiently small, thus partially establishing a De Giorgi conjecture in dimension one and codimension one. We also extend our result to a suitable anisotropic setting.
math.AP↗
On jump minimizing liftings for $\mathbb S^1$-valued maps and connections with Ambrosio-Tortorelli-type $Γ$-limits
This paper is concerned with the $Γ$-limits of Ambrosio-Tortorelli-type functionals, for maps $u$ defined on an open bounded set $Ω\subset\mathbb R^n$ and taking values in the unit circle $\mathbb S^1\subset\mathbb R^2$. Depending on the domain of the functional, two different $Γ$-limits are possible, one of which is nonlocal, and related to the notion of jump minimizing lifting, i.e., a lifting of a map $u$ whose measure of the jump set is minimal. The latter requires ad hoc compactness results for sequences of liftings which, besides being interesting by themselves, also allow to deduce existence of a jump minimizing lifting.
math.AP↗
On unique decomposition of knotted handlebodies
The paper considers the uniqueness question of factorization of a knotted handlebody in the $3$-sphere along decomposing $2$-spheres. We obtain a uniqueness result for factorization along decomposing $2$-spheres meeting the handlebody at three parallel disks. The result is used to examine handlebody-knot symmetry; particularly, the chirality of $6_{10}$ in the handlebody-knot table, previously unknown, is determined. In addition, an infinite family of hyperbolic handlebody-knots with homeomorphic exteriors is constructed.
math.GT↗
Relaxation of the area of the vortex map: a non-parametric Plateau problem for a catenoid containing a segment
Motivated by the study of the non-parametric area $\mathcal A$ of the graph of the vortex map $u$ (a two-codimensional singular surface in $\mathbb R^4$) over the disc $Ω\subset \mathbb R^2$ of radius $l$, we perform a careful analysis of the singular part of the relaxation of $\mathcal A$ computed at $u$. The precise description is given in terms of a area-minimizing surface in a vertical copy of $\mathbb R^3 \subset \mathbb R^4$, which is a sort of ``catenoid'' containing a segment corresponding to a radius of $Ω$. The problem involves an area-minimization with a free boundary part; several boundary regularity properties of the minimizer are inspected.
math.AP↗
The $L^1$-relaxed area of the graph of the vortex map: optimal upper bound
We compute an upper bound for the value of the $L^1$-relaxed area of the graph of the vortex map $u : B_l(0)\subset \mathbb R^2 \to \mathbb R^2$, $u(x):= x/\vert x\vert$, $x \neq 0$, for all values of $l>0$. Together with a previously proven lower bound, this upper bound turns out to be optimal. Interestingly, for the radius $l$ in a certain range, in particular $l$ not too large, a Plateau-type problem, having as solution a sort of catenoid constrained to contain a segment, has to be solved.
math.AP↗
Minimizing movements for the generalized power mean curvature flow
Motivated by a conjecture of De Giorgi, we consider the Almgren-Taylor-Wang scheme for mean curvature flow, where the volume penalization is replaced by a term of the form \[ \int_{EΔF} f\Big(\frac{ {\rm d}_F }τ\Big)~dx \] for $f$ ranging in a large class of strictly increasing continuous functions. In particular, our analysis covers the case \[ f(r) = r^α, \qquad r \geq 0, \quad α>0, \] considered by De Giorgi. We show that the generalized minimizing movement scheme converges to the geometric evolution equation \[ f(v) = - κ\quad \text{on $\partial E(t)$,} \] where $\{E(t)\}$ are evolving subsets of $\mathbb{R}^n,$ $v$ is the normal velocity of $\partial E(t),$ and $κ$ is the mean curvature of $\partial E(t)$. We extend our analysis to the anisotropic setting, and in the presence of a driving force. We also show that minimizing movements coincide with the smooth classical solution as long as the latter exists. Finally, we prove that in the absence of forcing, mean convexity and convexity are preserved by the weak flow.
math.AP↗
Crystalline hexagonal curvature flow of networks: short-time, long-time and self-similar evolutions
We study the crystalline curvature flow of planar networks with a single hexagonal anisotropy. After proving the local existence of a classical solution for a rather large class of initial conditions, we classify the homothetically shrinking solutions having one bounded component. We also provide an example of network shrinking to a segment with multiplicity two.
math.AP↗
Upper bounds for the relaxed area of $\mathbb S^1$-valued Sobolev maps and its countably subadditive interior envelope
Given a bounded open connected Lipschitz set $Ω\subset \mathbb R^2$, we show that the relaxed Cartesian area functional $\overline{\mathcal A}(u,Ω)$ of a map $u\in W^{1,1}(Ω;\mathbb S^1)$ is finite, and provide a useful upper bound for its value. Using this estimate, we prove a modified version of a De Giorgi conjecture [17] adapted to $W^{1,1}(Ω;\mathbb S^1)$, on the largest countably subadditive set function $\overline {\overline{\mathcal A}}(u, \cdot)$ smaller than or equal to $\overline{\mathcal A}(u,\cdot)$.
math.AP↗
Relaxed area of graphs of piecewise Lipschitz maps in the strict $BV$-convergence
We compute the relaxed Cartesian area in the strict $BV$-convergence on a class of piecewise Lipschitz maps from the plane to the plane, having jump made of several curves allowed to meet at a finite number of junction points. We show that the domain of this relaxed area is strictly contained in the domain of the classical $L^1$-relaxed area.
math.CA↗
The relaxed area of $\mathcal{S}^1$-valued singular maps in the strict $BV$-convergence
Given a bounded open set $Ω\subset \mathbb{R}^2$, we study the relaxation of the nonparametric area functional in the strict topology in $BV(Ω;\mathbb{R}^2)$, and compute it for vortex-type maps, and more generally for maps in $W^{1,1}(Ω;\mathcal{S}^1)$ having a finite number of topological singularities. We also extend the analysis to some specific piecewise constant maps in $BV(Ω;\mathcal{S}^1)$, including the symmetric triple junction map.
math.CA↗
Some aspects of anisotropic curvature flow of planar partitions
We consider the geometric evolution of a network in the plane, flowing by anisotropic curvature. We discuss local existence of a classical solution in the presence of several smooth anisotropies. Next, we discuss some aspects of the polycrystalline case.
math.DG↗