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.
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arXiv subjects
Maurizio Paolini
Publications and source records attributed to Maurizio Paolini.
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.
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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.
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A free boundary singular transport equation as a formal limit of a discrete dynamical system
We study the continuous version of a hyperbolic rescaling of a discrete game, called open mancala. The resulting PDE turns out to be a singular transport equation, with a forcing term taking values in $\{0,1\}$, and discontinuous in the solution itself. We prove existence and uniqueness of a certain formulation of the problem, based on a nonlocal equation satisfied by the free boundary dividing the region where the forcing is one (active region) and the region where there is no forcing (tail region). Several examples, most notably the Riemann problem, are provided, related to singularity formation. Interestingly, the solution can be obtained by a suitable vertical rearrangement of a multi-function. Furthermore, the PDE admits a Lyapunov functional.
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On closed oriented surfaces in the 3-sphere
In this paper we study embeddings of oriented connected closed surfaces in $\mathbb S^3$. We define a complete invariant, the fundamental span, for such embeddings, generalizing the notion of the peripheral system of a knot group. From the fundamental span, several computable invariants are derived and employed to study handlebody knots, bi-knotted surfaces, and chirality of knots. These invariants are capable to distinguish inequivalent handlebody knots and bi-knotted surfaces with homeomorphic complements. Particularly, we obtain an alternative proof of the inequivalence of Ishii et al.'s handlebody knots $5_{1}$ and $6_{4}$, and also construct an infinite family of pairs of inequivalent bi-knotted surfaces with homeomorphic complements. An interpretation of Fox's invariant in terms of the fundamental span is discussed and used to show $9_{42}$ and $10_{71}$ in the Rolfsen knot table are chiral; their chirality is known to be undetectable by the Jones and HOMFLY-PT polynomials.
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A complete invariant for closed surfaces in the three-sphere
Associated to an embedded surface in the $3$-sphere, we construct a diagram of fundamental groups, and prove that it is a complete invariant, wherefrom we deduce complete invariants of handlebody links, tunnels of handlebody links, and spatial graphs.The main ingredients in the proof of the completeness are a generalization of the Kneser conjecture for $3$-manifolds with boundary proved also here, and extensions of Waldhausen's theorem by Evans, Tucker and Swarup. Computable invariants of handlebody links derived therefrom are calculated.
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A table of $n$-component handlebody links of genus $n+1$ up to six crossings
A handlebody link is a union of handlebodies of positive genus embedded in 3-space, which generalizes the notion of links in classical knot theory. In this paper, we consider handlebody links with one genus 2 handlebody and $n-1$ solid tori, $n>1$. Our main result is the complete classification of such handlebody links with six crossings or less, up to ambient isotopy.
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Numerical irreducibility criteria for handlebody links
In this paper we define a set of numerical criteria for a handlebody link to be irreducible. It provides an effective, easy-to-implement method to determine the irreducibility of handlebody links; particularly, it recognizes the irreducibility of all handlebody knots in the Ishii-Kishimoto-Moriuchi-Suzuki knot table and most handlebody links in the Bellettini-Paolini-Paolini-Wang link table.
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On the relaxed area of the graph of discontinuous maps from the plane to the plane taking three values with no symmetry assumptions
In this paper we estimate from above the area of the graph of a singular map $u$ taking a disk to three vectors, the vertices of a triangle, and jumping along three $\mathcal{C}^2-$ embedded curves that meet transversely at only one point of the disk. We show that the relaxed area can be estimated from above by the solution of a Plateau-type problem involving three entangled nonparametric area-minimizing surfaces. The idea is to "fill the hole" in the graph of the singular map with a sequence of approximating smooth two-codimensional surfaces of graph-type, by imagining three minimal surfaces, placed vertically over the jump of $u$, coupled together via a triple point in the target triangle. Such a construction depends on the choice of a target triple point, and on a connection passing through it, which dictate the boundary condition for the three minimal surfaces. We show that the singular part of the relaxed area of $u$ cannot be larger than what we obtain by minimizing over all possible target triple points and all corresponding connections.
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Covers, soap films and BV functions
In this paper we first review the covering space method with constrained BV functions for solving the classical Plateau's problem. Next, we carefully analyze some interesting examples of soap films compatible with the covering space method: in particular, the case of a soap film only partially wetting a space curve, a soap film spanning a cubical frame but having a large tunnel, aa soap film that retracts onto its boundary, hence not modelable with the Reifenberg method, and various soap films spanning an octahedral frame.
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Triple covers and a non-simply connected surface spanning an elongated tetrahedron and beating the cone
By using a suitable triple cover we show how to possibly model the construction of a minimal surface with positive genus spanning all six edges of a tetrahedron, working in the space of BV functions and interpreting the film as the boundary of a Caccioppoli set in the covering space. After a question raised by R. Hardt in the late 1980's, it seems common opinion that an area-minimizing surface of this sort does not exist for a regular tetrahedron, although a proof of this fact is still missing. In this paper we show that there exists a surface of positive genus spanning the boundary of an elongated tetrahedron and having area strictly less than the area of the conic surface.
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Exploring the "Rubik's Magic" universe
By using two different invariants for the Rubik's Magic puzzle, one of metric type, the other of topological type, we can dramatically reduce the universe of constructible configurations of the puzzle. Finding the set of actually constructible shapes remains however a challenging task, that we tackle by first reducing the target shapes to specific configurations: the octominoid 3D shapes, with all tiles parallel to one coordinate plane; and the planar "face-up" shapes, with all tiles (considered of infinitesimal width) lying in a common plane and without superposed consecutive tiles. There are still plenty of interesting configurations that do not belong to either of these two collections. The set of constructible configurations (those that can be obtained by manipulation of the undecorated puzzle from the starting situation) is a subset of the set of configurations with vanishing invariants. We were able to actually construct all octominoid shapes with vanishing invariants and most of the planar "face-up" configurations. Particularly important is the topological invariant, of which we recently found mention in a paper by Tom Verhoeff.
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An optimal bound on the number of moves for open Mancala
We determine the optimal bound for the maximum number of moves required to reach a periodic configuration of open mancala (also called open owari), inspired by a popular African game. A mancala move can be interpreted as a map from the set of compositions of a given integer in itself, thus relating our result to the study of the corresponding finite dynamical system.
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On the area of the graph of a piecewise smooth map from the plane to the plane with a curve discontinuity
In this paper we provide an estimate from above for the value of the relaxed area functional for a map defined on a bounded domain of the plane with values in the plane and discontinuous on a regular simple curve with two endpoints. We show that, under suitable assumptions, the relaxed area does not exceed the area of the regular part of the map, with the addition of a singular term measuring the area of a disk type solution of the Plateau's problem spanning the two traces of the map on the jump. The result is valid also when the area minimizing surface has self intersections. A key element in our argument is to show the existence of what we call a semicartesian parametrization of this surface, namely a conformal parametrization defined on a suitable parameter space, which is the identity in the first component. To prove our result, various tools of parametric minimal surface theory are used, as well as some result from Morse theory.
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