Counting Weighted Bi-Colored Plane Trees and Their Geometric Applications

Sicheng Lu; Yi Song

Home/Archive/arXiv

arXiv · 2606.21074

Counting Weighted Bi-Colored Plane Trees and Their Geometric Applications

Sicheng Lu·Yi Song

View original source ↗Read source document ↗Export metadata ↓

Abstract

This work solves the enumeration problem for weighted bi-colored plane trees with prescribed numbers of black and white vertices, together with prescribed total edge weights at each vertex. An exact closed formula for a particular case is obtained, and a unified algorithmic method for the general case is provided. We then apply this result to two geometric problems. Firstly, we compute the strong Hurwitz number for a special class of branch datum between Riemann spheres with three branched points. This is done by counting the dessins d'enfants that faithfully record such branched covers, and a clear correspondence between dessins and weighted trees in such case. Secondly, we study the geometric moduli space for a special class of extremal Kähler metrics on Riemann sphere (HCMU spheres), with a single conical singularity. We classify and enumerate the connected components of the moduli space with respect to the Gromov-Hausdorff topology. This is based on an efficient representation of these metric surfaces, and a study of Gromov-Hausdorff limits of such surfaces.

Explore related subjects

math.GT math.CO math.CV math.DG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Sicheng Lu, Yi Song. 2026-07-21. Counting Weighted Bi-Colored Plane Trees and Their Geometric Applications. https://arxiv.org/abs/2606.21074

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Hyperbolic links associated to Hamiltonian subgraphs in simple $3$-polytopes

We build a large family of hyperbolic links with an explicit decomposition of the complement into right-angled hyperbolic polytopes of finite volume. Namely, in a series of papers A.D.Mednykn and A.Yu.Vesnin introduced a construction that for a given right-angled polytope $P$ in geometry $\mathbb L^3$, $\mathbb R^3$, $\mathbb S^3$, $\mathbb L^2\times \mathbb R$, $\mathbb S^2\times \mathbb R$ and a Hamiltonian cycle, theta-subgraph or $K_4$-subgraph $Γ$ in the $1$-skeleton of $P$ builds a geometric $3$-manifold $N(P,Γ)$ with an involution $τ$ such that $N(P,Γ)/\langleτ\rangle\simeq S^3$. The brach set of the corresponding $2$-sheeted branched covering $N(P,Γ)\to S^3$ is a link $C_Γ\subset S^3$ consisting of trivially embedded circles. This construction reformulated in the language of toric topology works for such a subgraph $Γ$ in any simple $3$-polytope $P$ and gives a topological $3$-manifold $N(P,Γ)$. We give a criterion when $S^3\setminus C_Γ$ has a complete hyperbolic structure of finite volume and generalize this criterion to similar links in $3$-manifolds different from $S^3$. We prove that hyperbolic links $C_Γ$ are parametrized by nonselfcrossing Eulerian cycles, Eulerian theta-subgraphs and Eulerian $K_4$-subgraphs in hyperbolic right-angled $3$-polytopes of finite volume in $\mathbb L^3$ with $0$, $2$ or $4$ finite vertices. The complement $S^3\setminus C_Γ$ is glued of $4$, $8$ or $16$ copies of the corresponding right-angled polytope. We give a criterion when the link $C_Γ$ consists of mutually unlinked circles and prove that if such a link is nontrivial, then it contains the Borromean rings. The latter problem is motivated by the Efimov effect in quantum mechanics.

math.GT↗

arXiv2026

Knots and the Sierpinski Tetrahedron

In this paper we prove that there are infinitely many knots that cannot be embedded in the 1-skeletons of the finite iterations of the Sierpinski tetrahedron fractal. We do this by proving that such an embedding induces a sphere decomposition of weight at most 6. There are infinitely many knots with spherewidth greater than this.