arXiv subjects
Publications and source records attributed to Haimiao Chen.
We apply Dijkgraaf-Witten invariant over an semiproduct of abelian groups to show that, if the $k/\ell$-surgery along a knot $K$ results in a small Seifert 3-manifold with multiplicities $a_1,a_2,a_3$, then many constraints on $k,a_1,a_2,a_3$ can be read off from the Alexander polynomial of $K$.
We completely determine finite abelian regular branched covers of the 2-sphere $S^2$ with the property that each homeomorphism of $S^2$ preserving the branching set can be lifted.
We briefly review 3-dimensional untwisted Dijkgraaf-Witten theory over a finite group $Γ$, and present a method of computing untwisted Dijkgraaf-Witten invariants for arborescent links. Some explicit formulas are given when $Γ=\mathbb{Z}/p\mathbb{Z}\rtimes\mathbb{Z}/(p-1)\mathbb{Z}$ for an odd prime $p$.
For each Montesinos knot $K$, we propose an efficient method to explicitly determine the irreducible ${\rm SL}(2,\mathbb{C})$-character variety, and show that it can be decomposed as $\mathcal{X}_0(K)\sqcup\mathcal{X}_1(K)\sqcup\mathcal{X}_2(K)\sqcup\mathcal{X}'(K)$, where $\mathcal{X}_0(K)$ consists of trace-free characters, $\mathcal{X}_1(K)$ consists of characters of "unions" of representations of rational knots (or rational link, which appears at most once), $\mathcal{X}_2(K)$ is an algebraic curve, and $\mathcal{X}'(K)$ consists of finitely many points when $K$ satisfies a generic condition.
We present an efficient method for computing the ${\rm SL}(3,\mathbb{C})$-character varieties of two-generator groups.
For any integers $m,n$ with $m\ne 0$ and $n>0$, let $G_{m,n}$ denote the group presented by $\langle x,y,z\mid x=[z^m,x][z^n,y]\rangle$; for any integers $m,n>0$, let $H_{m,n}$ denote the group presented by $\langle x,y,z\mid x=[x^m,z^n][y,z]\rangle$. By investigating cohomology jump loci of irreducible ${\rm GL}(2,\mathbb{C})$-character varieties, we show: if $m,m'\ne 0$, $n,n'>0$ and $G_{m',n'}\cong G_{m,n}$, then $m=m',n=n'$; if $m,m',n,n'>0$ and $H_{m',n'}\cong H_{m,n}$, then $m'=m, n'=n$.
In recent years, twisted Alexander polynomial has been playing an important role in low-dimensional topology. For Montesinos links, we develop an efficient method to compute the twisted Alexander polynomial associated to any linear representation. In particular, formulas for multi-variable Alexander polynomials of these links can be easily derived.
Suppose $Γ$ is a discrete group, and $α\in Z^3(BΓ;A)$, with $A$ an abelian group. Given a representation $ρ:π_1(M)\toΓ$, with $M$ a closed 3-manifold, put $F(M,ρ)=\langle(Bρ)^\ast[α],[M]\rangle$, where $Bρ:M\to BΓ$ is a continuous map inducing $ρ$ which is unique up to homotopy, and $\langle-,-\rangle:H^3(M;A)\times H_3(M;\mathbb{Z})\to A$ is the pairing. We extend the definition of $F(M,ρ)$ to manifolds with corners, and establish a gluing law. Based on these, we present a practical method for computing $F(M,ρ)$ when $M$ is given by a surgery along a link $L\subset S^3$. In particular, the Chern-Simons invariant can be computed this way.
For each even classical pretzel knot $P(2k_1+1,2k_2+1,2k_3)$, we determine the character variety of irreducible ${\rm SL}(2,\mathbb{C})$-representations, and clarify the steps of computing its A-polynomial.
Given a link $L\subset S^3$, a representation $π_1(S^3-L)\to{\rm SL}(2,\mathbb{C})$ is {\it trace-free} if it sends each meridian to an element with trace zero. We present a method for completely determining trace-free ${\rm SL}(2,\mathbb{C})$-representations for arborescent links. Concrete computations are done for a class of 3-bridge arborescent links.
We determine the ${\rm SL}(2,\mathbb{C})$-character variety for each odd classical pretzel knot $P(2k_1+1,2k_2+1,2k_3+1)$, and present a method for computing its A-polynomial.
Given a link $L$, a representation $π_1(S^{3}-L)\to {\rm SL}(2,\mathbb{C})$ is {\it tracefree} if the image of each meridian has trace zero. We determine the conjugacy classes of tracefree representations when $L$ is a Montesinos link.
It is shown that, if a link $\tilde{L}\subset S^3$ is $p^k$-periodic with $p$ prime and $k\ge 1$, and $L$ is the quotient link, then the groups of $\tilde{L}$ and $L$ can be related by counting homomorphisms to any finite group $Γ$ whose order is not divisible by $p$.
For each odd prime power $q$ with $q\ge 5$ and $4\mid q-1$, we investigate the structure of the representation category of the quantum double of ${\rm SL}(2,q)$, determining its tensor products and braidings.
A metacyclic group $H$ can be presented as $\langle α,β\mid α^{n}=1, \ β^{m}=α^{t}, \ βαβ^{-1}=α^{r}\rangle$ for some $n,m,t,r$. Each endomorphism $σ$ of $H$ is determined by $σ(α)=α^{x_{1}}β^{y_{1}}, σ(β)=α^{x_{2}}β^{y_{2}}$ for some integers $x_{1},x_{2},y_{1},y_{2}$. We give sufficient and necessary conditions on $x_{1},x_{2},y_{1},y_{2}$ for $σ$ to be an automorphism.
We determine the structure of automorphism group or each nonsplit metacyclic 2-group. This completes the work on automorphism groups of metacyclic $p$-groups.
Given a finite group $Γ$, a regular $t$-balanced Cayley map (RBCM$_{t}$ for short) is a regular Cayley map $\mathcal{CM}(G,Ω,ρ)$ such that $ρ(ω)^{-1}=ρ^{t}(ω)$ for all $ω\inΩ$. In this paper, we clarify a connection between quotients of polynomial rings and RBCM$_{t}$'s on abelian groups, so as to propose a new approach for classifying RBCM$_{t}$'s. We obtain many new results, in particular, a complete classification for RBCM$_{t}$'s on abelian 2-groups.
We derive a formula for the Dijkgraaf-Witten invariants of orientable Seifert 3-manifolds with orientable bases.