Benjamin Bode

Benjamin Bode
Spanish National Research Council | CSIC · Instituto de Ciencias Matemáticas

PhD

About

17
Publications
1,443
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94
Citations
Introduction
I'm working on knots and links and their relations to other branches of mathematics and physics. Examples are constructions of real polynomial maps with isolated singularities leading to real algebraic links, or the study of knots in different physical systems such as knotted field lines in electromagnetic fields. My homepage is https://sites.google.com/my.bristol.ac.uk/benjaminbode/

Publications

Publications (17)
Preprint
We study four (a priori) different ways in which an open book decomposition of the 3-sphere can be braided. These include generalised exchangeability defined by Morton and Rampichini and mutual braiding defined by Rudolph, which were shown to be equivalent by Rampichini, as well as P-fiberedness and a property related to simple branched covers of $...
Article
Full-text available
Persistent topological structures in physical systems have become increasingly important over the last years. Electromagnetic fields with knotted field lines play a special role among these, since they can be used to transfer their knottedness to other systems like plasmas and quantum fluids. In null electromagnetic fields the electric and the magn...
Preprint
In null electromagnetic fields the electric and the magnetic field lines evolve like unbreakable elastic filaments in a fluid flow. In particular, their topology is preserved for all time. We prove that for every link $L$ there is such an electromagnetic field that satisfies Maxwell's equations in free space and that has closed electric and magneti...
Article
For every link L we construct a complex algebraic plane curve that intersects S3 transversally in a link L˜ that contains L as a sublink. This construction proves that every link L is the sublink of a quasipositive link that is a satellite of the Hopf link. The explicit construction of the corresponding complex polynomial f:C2→C can be used to give...
Preprint
Full-text available
A geometric braid $B$ can be interpreted as a loop in the space of monic complex polynomials with distinct roots. This loop defines a function $g:\mathbb{C}\times S^1\to\mathbb{C}$ that vanishes on $B$. We define the set of P-fibered braids as those braids that can be represented by loops of polynomials such that the corresponding function $g$ indu...
Article
We construct an infinite tower of covering spaces over the configuration space of [Formula: see text] distinct nonzero points in the complex plane. This results in an action of the braid group [Formula: see text] on the set of [Formula: see text]-adic integers [Formula: see text] for all natural numbers [Formula: see text]. We study some of the pro...
Conference Paper
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Preprint
Full-text available
We present an algorithm that takes as input any element $B$ of the loop braid group and constructs a polynomial $f:\mathbb{R}^5\to\mathbb{R}^2$ such that the intersection of the vanishing set of $f$ and the unit 4-sphere contains the closure of $B$. The polynomials can be used to create real analytic time-dependent vector fields with zero divergenc...
Preprint
For every link $L$ we construct a complex algebraic plane curve that intersects $S^3$ transversally in a link $\tilde{L}$ that contains $L$ as a sublink. This construction proves that every link $L$ is the sublink of a quasipositive link that is a satellite of the Hopf link. The explicit construction of the complex plane curve can be used to give u...
Preprint
We construct an infinite tower of covering spaces over the configuration space of $n-1$ distinct non-zero points in the complex plane. This results in an action of the affine braid group $\mathbb{B}_{n-1}^{aff}$ on the $n$-adic integers $\mathbb{Z}_n$ for all natural numbers $n\geq 2$. With similar constructions we obtain several actions of the Art...
Article
We study the minimal crossing number $c(K_{1}\# K_{2})$ of composite knots $K_{1}\# K_{2}$, where $K_1$ and $K_2$ are prime, by relating it to the minimal crossing number of spatial graphs, in particular the $2n$-theta curve $\theta_{K_{1},K_{2}}^n$ that results from tying $n$ of the edges of the planar embedding of the $2n$-theta graph into $K_1$...
Article
We show that if a braid $B$ can be parametrised in a certain way, then previous work can be extended to a construction of a polynomial $f:\mathbb{R}^4\to\mathbb{R}^2$ with the closure of $B$ as the link of an isolated singularity of $f$, showing that the closure of $B$ is real algebraic. In particular, we prove that closures of squares of strictly...
Article
Full-text available
We describe a procedure that creates an explicit complex-valued polynomial function of three-dimensional space, whose nodal lines are the three-twist knot $5_2$. The construction generalizes a similar approach for lemniscate knots: a braid representation is engineered from finite Fourier series and then considered as the nodal set of a certain comp...
Article
Full-text available
We describe an algorithm that for every given braid [Formula: see text] explicitly constructs a function [Formula: see text] such that [Formula: see text] is a polynomial in [Formula: see text], [Formula: see text] and [Formula: see text] and the zero level set of [Formula: see text] on the unit three-sphere is the closure of [Formula: see text]. T...
Article
Full-text available
We give an explicit construction of complex maps whose nodal line have the form of lemniscate knots. We review the properties of lemniscate knots, defined as closures of braids where all strands follow the same transverse (1, $\ell$) Lissajous figure, and are therefore a subfamily of spiral knots generalising the torus knots. We then prove that suc...

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Project (1)
Project
Aim to understand the way knot topology manifests in different physical settings, including quantum mechanics, optics, particle physics and molecular biology.