## About

30

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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/view/benjaminbode/

**Skills and Expertise**

## Publications

Publications (30)

The curves of zero intensity of a complex optical field can form knots and links: optical vortex knots. Both theoretical constructions and experiments have so far been restricted to the very small families of torus knots or lemniscate knots. Here we describe a mathematical construction that presumably allows us to generate optical vortices in the s...

We introduce the notion of a (strongly) inner non-degenerate mixed function \(f:{\mathbb {C}}^2\rightarrow {\mathbb {C}}.\) We show that inner non-degenerate mixed polynomials have weakly isolated singularities and strongly inner non-degenerate mixed polynomials have isolated singularities. Furthermore, under one additional assumption, which we cal...

Let $$g_t$$ g t be a loop in the space of monic complex polynomials in one variable of fixed degree n . If the roots of $$g_t$$ g t are distinct for all t , they form a braid $$B_1$$ B 1 on n strands. Likewise, if the critical points of $$g_t$$ g t are distinct for all t , they form a braid $$B_2$$ B 2 on $$n-1$$ n - 1 strands. In this paper we stu...

Let $g_t$ be a loop in the space of monic complex polynomials in one variable of fixed degree $n$. If the roots of $g_t$ are distinct for all $t$, they form a braid $B_1$ on $n$ strands. Likewise, if the critical points of $g_t$ are distinct for all $t$, they form a braid $B_2$ on $n-1$ strands. In this paper we study the relationship between $B_1$...

Semiholomorphic polynomials are functions f:C2→C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:{\mathbb {C}}^2\rightarrow {\mathbb {C}}$$\end{document} that can be w...

Let $f:\mathbb{C}^2\to\mathbb{C}$ be an inner non-degenerate mixed polynomial with a nice Newton boundary with $N$ compact 1-faces. In the first part of this series of papers we showed that $f$ has a weakly isolated singularity and that its link can be constructed from a sequence of links $L_1, L_2,\ldots,L_N$, each of which is associated with a co...

Mixed polynomials $f:\mathbb{C}^2\to\mathbb{C}$ are polynomial maps in complex variables $u$ and $v$ as well as their complex conjugates $\bar{u}$ and $\bar{v}$. They are therefore identical to the set of real polynomial maps from $\mathbb{R}^4$ to $\mathbb{R}^2$. We generalize Mondal's notion of partial non-degeneracy from holomorphic polynomials...

We study four (a priori) different ways in which an open book decomposition of the 3-sphere can be defined to 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 branch...

Null solutions to Maxwell's equations in free space have the property that the topology of the electric and magnetic lines is preserved for all time. In this article we connect the study of a particularly relevant class of null solutions (related to the Hopf fibration) with the existence of pairs of volume preserving Legendrian fields with respect...

A link in $S^3$ is called real algebraic if it is the link of an isolated singularity of a polynomial map from $\mathbb{R}^4$ to $\mathbb{R}^2$. It is known that every real algebraic link is fibered and it is conjectured that the converse is also true. We prove this conjecture for a large family of fibered links, which includes closures of T-homoge...

Semiholomorphic polynomials are functions $f:\mathbb{C}^2\to\mathbb{C}$ that can be written as polynomials in complex variables $u$, $v$ and the complex conjugate $\overline{v}$. We prove the semiholomorphic analogoue of Akbulut's and King's "All knots are algebraic", that is, every link type in the 3-sphere arises as the link of a weakly isolated...

We introduce the notion of a (strongly) inner non-degenerate mixed function $f:\mathbb{C}^2\to\mathbb{C}$. We show that inner non-degenerate mixed polynomials have weakly isolated singularities and strongly inner non-degenerate mixed polynomials have isolated singularities. Furthermore, under one additional assumption, which we call "niceness", the...

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 $...

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...

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...

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...

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...

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...

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...

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...

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...

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 $K_1$...

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...

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...

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...

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...