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Haimiao Chen

Publications and source records attributed to Haimiao Chen.

At least 19 recordsLinked to original sources
arXiv2026

Presentations of skein algebras of genus 2

We give a presentation for the Kauffman bracket skein algebra of a closed or one-holed oriented surface of genus $2$. This is the first such result for genus greater than $1$.

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arXiv2026

On skein algebras of planar surfaces

Let $R$ be a commutative ring with identity and a fixed invertible element $q^{\frac{1}{2}}$. Let $\mathcal{S}_n$ denote the Kauffman bracket skein algebra of the $n$-holed disk $Σ_{0,n+1}$ over $R$. When $q+q^{-1}$ is invertible, in 2000 Przytycki and Sikora found a set of $n+{n\choose 2}+{n\choose 3}$ generators for $\mathcal{S}_n$; we show that the ideal of defining relations among these generators is generated by relations of degree $\le6$ supported by certain subsurfaces diffeomorphic to $Σ_{0,k+1}$ with $k\le 6$. When $q+q^{-1}$ is not invertible, a set of $2^n-1$ generators for $\mathcal{S}_n$ was known to Bullock in 1999; we show that the ideal of defining relations is generated by relations of degree $\le 2k+2$ supported by certain subsurfaces diffeomorphic to $Σ_{0,k+1}$ with $k\le n$. These results are substantial progresses towards answering Problem 1.92 (J) in the Kirby's list.

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arXiv2026

Computing Alexander polynomials for arborescent links

Alexander polynomial was born one century ago, but explicit formulas have been found for only a few families of links. In this paper, we present an efficient method of computing Alexander polynomial for arborescent links. Applying this method, we express the Alexander polynomials of Montesinos links in terms of certain polynomials associated to rational tangles which can be computed recursively. Specifically, we deduce explicit closed formulas for all pretzel links.

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arXiv2026

High-dimensional components of ${\rm SL}(2,\mathbb{C})$-character varieties of prime knots

For a prime knot $K$, we give sufficient conditions for the existence of a component $\mathcal{C}$ of the irreducible ${\rm SL}(2,\mathbb{C})$-character variety of $K$ with $\dim\mathcal{C}>1$, and give a lower bound for $\dim\mathcal{C}$. Specifically, we improve a result of Paoluzzi and Porti on Montesinos knots, and positively answer a question posed by Culler and Dunfield in 2018.

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arXiv2025

Torsion in Kauffman bracket skein module of a $4$-strand Montesinos knot exterior

For an oriented $3$-manifold $M$, let $\mathcal{S}(M)$ denote its Kauffman bracket skein module over $\mathbb{Z}[q^{\pm\frac{1}{2}}]$. We show that $\mathcal{S}(M)$ admits torsion when $M$ is the exterior of the Montesinos knot $K(a_1/b_1,a_2/b_2,a_3/b_4,a_4/b_4)$ with each $b_i\ge 3$. This provides a negative answer to Problem 1.92 (G)-(i) in the Kirby's list, which asks whether $\mathcal{S}(M)$ is free when $M$ is irreducible and has no incompressible non-boundary parallel torus.

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arXiv2025

Finite $p$-groups of class $2$ as central extensions

Finite $p$-groups of nilpotency class 2 are treated from the perspective of central extensions. Given finite abelian groups $G,A$, we derive an explicit formula for cocycles representing elements of $H^2(G,A)$, compute $H^2(G,A)$, and describe the actions of ${\rm End}(G)$ and ${\rm End}(A)$ on $H^2(G,A)$. These are used to provide an efficient criterion for lifting endomorphisms of $G$ to homomorphisms between two central extensions. Subsequently, we present two applications to illustrate the usefulness of this approach, in the case $p>2$. First, we recover the classification of two-generator $p$-groups of class $2$ up to isomorphism, and compute the order of the automorphism group for each isomorphism class. Second, we construct a family of nonabelian $p$-groups of order $p^7$ whose automorphism groups are abelian.

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arXiv2025

The ${\rm SL}(2,\mathbb{C})$-character variety of $8_{18}$

The knot $8_{18}$ is the first non-arborescent hyperbolic knot. In 2020, Paoluzzi and Porti found its ${\rm SL}(2,\mathbb{C})$-character variety with the aid of a computer, but many details were omitted. In this paper, we determine the character variety by a software-free procedure, which is easy to follow and enlightening. Along the way, we develop an efficient method for working with simultaneous conjugacy classes of four elements of ${\rm SL}(2,\mathbb{C})$.

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arXiv2025

Kauffman bracket skein algebra of the 4-holed disk

We give a monomial basis for the Kauffman bracket skein algebra of the $4$-holed disk, and find a presentation. This is based on an insight into the ${\rm SL}(2,\mathbb{C})$-character variety of the rank $4$ free group.

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arXiv2025

Kauffman bracket skein module of the $(3,3,3,3)$-pretzel link exterior

We show that the Kauffman bracket skein module of the $(3,3,3,3)$-pretzel link exterior over $\mathbb{Q}(q^{\frac{1}{2}})$ is not finitely generated as a module over $\mathbb{Q}(q^{\frac{1}{2}})[t_1,t_2]$, where $t_1,t_2$ are the meridians of two components. This disproves a finiteness conjecture of Detcherry proposed in 2021.

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arXiv2024

On the structure of Kauffman bracket skein algebra of a surface

Suppose $R$ is a commutative ring with identity and a fixed invertible element $q^{\frac{1}{2}}$ such that $q+q^{-1}$ is invertible. For an oriented surface $Σ$, let $\mathcal{S}(Σ;R)$ denote the Kauffman bracket skein algebra of $Σ$ over $R$. It is shown that to each embedded graph $G\subsetΣ$ satisfying that $Σ\setminus G$ is homeomorphic to a disk and some other mild conditions, one can associate a generating set for $\mathcal{S}(Σ;R)$, and the ideal of defining relations is generated by relations of degree at most $6$ supported by certain small subsurfaces.

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arXiv2024

A new approach to the genus spectra of abelian $p$-groups

Given a finite group $G$, the {\it genus spetrum} ${\rm sp}(G)$ of $G$ is the set of integers $g\geq 0$ such that $G$ can act faithfully on an orientable closed surface of genus $g$ by orientation-preserving homeomorphisms. The determination of ${\rm sp}(G)$ is a classical topic and has a long history, but progress is lacked. In this paper, when $G$ is an abelian $p$-group with $p>2$, we propose a new approach to ${\rm sp}(G)$, giving a structural description for ${\rm sp}(G)$ in terms of a function which can be computed in finitely many steps.

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arXiv2024

Regular $t$-balanced Cayley maps on split metacyclic $2$-groups

A regular $t$-balanced Cayley map on a group $Γ$ is an embedding of a Cayley graph on $Γ$ into a surface with certain special symmetric properties. We completely classify regular $t$-balanced Cayley maps for a class of split metacyclic $2$-groups.

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arXiv2023

Presentations of Kauffman bracket skein algebras of planar surfaces

Let $R$ be a commutative ring with identity and a fixed invertible element $q^{\frac{1}{2}}$, and suppose $q+q^{-1}$ is invertible in $R$. For each planar surface $Σ_{0,n+1}$, we present its Kauffman bracket skein algebra over $R$ by explicit generators and relations. The presentation is independent of $R$, and can be considered as a quantization of the trace algebra of $n$ generic $2\times 2$ unimodular matrices.

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arXiv2023

The ${\rm SL}(2,\mathbb{C})$-character variety of an arborescent knot

We describe a procedure for computing the ${\rm SL}(2,\mathbb{C})$-character variety of an arborescent knot. Along the way, we clarify several facts about representations of arborescent tangles. Then we study a family of hyperbolic knots whose exteriors contain closed essential surfaces, showing that each of these knots has $1$-dimensional character variety. This provides infinitely many positive answers to a question of Boyer and Zhang posed in 1998.

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