Related papers: Root laminations of arbitrary degree

Thurston introduced $\si_d$-invariant laminations (where $\si_d(z)$ coincides with $z^d:\ucirc\to \ucirc$, $d\ge 2$) and defined \emph{wandering $k$-gons} as sets $\T\subset \ucirc$ such that $\si_d^n(\T)$ consists of $k\ge 3$ distinct…

We associate to any endomorphism of the punctured affine space over some field an element in the Witt group of the base field that we call degree. We use this degree to give a counter-example to a question on unimodular rows

A degree-regular triangulation is one in which each vertex has identical degree. Our main result is that any such triangulation of a (possibly non-compact) surface $S$ is geometric, that is, it is combinatorially equivalent to a geodesic…

The degree matrix of a graph is the diagonal matrix with diagonal entries equal to the degrees of the vertices of $X$. If $X_1$ and $X_2$ are graphs with respective adjacency matrices $A_1$ and $A_2$ and degree matrices $D_1$ and $D_2$, we…

For a non-elementary subgroup of the mapping class group of a surface, we study its invariant Radon measures on the space of measured laminations, by classifying them on the recurrent measured laminations. In particular, given a…

To investigate the degree $d$ connectedness locus, Thur\-ston studied \emph{$\sigma_d$-invariant laminations}, where $\sigma_d$ is the $d$-tupling map on the unit circle, and built a topological model for the space of quadratic polynomials…

Conformal invariance plays a significant role in many areas of Physics, such as conformal field theory, renormalization theory, turbulence, general relativity. Naturally, it also plays an important role in geometry: theory of Riemannian…

We describe in this note a new invariant of rooted trees. We argue that the invariant is interesting on it own, and that it has connections to knot theory and homological algebra. However, the real reason that we propose this invariant to…

A proper $q$-coloring of a graph is an assignment of one of $q$ colors to each vertex of the graph so that adjacent vertices are colored differently. Sample uniformly among all proper $q$-colorings of a large discrete cube in the integer…

The loop graph of an infinite type surface is an infinite diameter hyperbolic graph first studied in detail by Juliette Bavard. An important open problem in the study of infinite type surfaces is to describe the boundary of the loop graph…

We consider the sum of dichromatic polynomials over non-separable rooted planar maps, an interesting special case of which is the enumeration of such maps. We present some known results and derive new ones. The general problem is equivalent…

The Tautological Lamination arises in holomorphic dynamics as a combinatorial model for the geometry of 1-dimensional slices of the Shift Locus. In each degree $q$ the tautological lamination defines an iterated sequence of partitions of…

We define additional gradings on two generalisations of Khovanov homology (one due to the first author, the other due to the second), and use them to define invariants of various kinds of embeddings. These include invariants of links in…

Let G be a simple complex algebraic group. By using a notion of a G-category we define invariants of tangles with flat G-connections in their complements. We also show that quantized universal enveloping algebras at roots of unity provide…

Following our previous work [18], we introduce the notions of partial seed homomorphisms and partial ideal rooted cluster morphisms. Related to the theory of Green's equivalences, the isomorphism classes of sub-rooted cluster algebras of a…

We consider a plane polynomial vector field $P(x,y)dx+Q(x,y)dy$ of degree $m>1$. To each algebraic invariant curve of such a field we associate a compact Riemann surface with the meromorphic differential $\omega=dx/P=dy/Q$. The asymptotic…

The combinatorial Mandelbrot set is a continuum in the plane, whose boundary can be defined, up to a homeomorphism, as the quotient space of the unit circle by an explicit equivalence relation. This equivalence relation was described by…

Invariant minimal surfaces in the real special linear group of degree 2 with canonical Riemannian and Lorentzian metrics are studied. Constant mean curvature surfaces with vertically harmonic Gau{\ss} map are classified.

We introduce the Plaque Topology on the inverse limit of a branched covering self-map of a Riemann surface of a finite degree greater than one. We present the notions of regular and irregular points in the setting of this Plaque Inverse…

We introduce a notion of topological quandle. Given a topological quandle $Q$ we associate to every classical link $L$ in $\R ^3$ an invariant $J_Q(L)$ which is a topological space (defined up to a homeomorphism). The space $J_Q(L)$ can be…