The non semi-simple TQFT of the sphere with four punctures

Almost simple geodesics on the triply punctured sphere

Every closed hyperbolic geodesic $γ$ on the triply--punctured sphere $M =\widehat{\mathbb C} - \{0,1,\infty\}$ has a self--intersection number $I(γ) \ge 1$ and a combinatorial length $L(γ) \ge 2$, the latter defined by the number of times $γ$ passes through the upper halfplane. In this paper we show that $δ(γ) = I(γ) - L(γ) \ge -1$ for all closed geodesics; and that for each fixed $δ$, the number of geodesics with invariants $(δ,L)$ is given exactly by a quadratic polynomial $p_δ(L)$ for all $L \ge 4 + δ$.

On pseudo-Anosov maps with small dilatations on punctured Riemann spheres

Let $S_n$ be a punctured Riemann spheres $\mathbf{S}^2\backslash \{x_1,..., x_n\}$. In this paper, we investigate pseudo-Anosov maps on $S_n$ that are isotopic to the identity on $S_n\cup \{x_n\}$ and have the smallest possible dilatations. We show that those maps cannot be obtained from Thurston's construction (that is the products of two Dehn twists). We also prove that those pseudo-Anosov maps $f$ on $S_n$ with the minimum dilatations can never define a trivial mapping class as any puncture $x_i$ of $S_n$ is filled in. The main tool is to give both lower and upper bounds estimations for dil…

Asymptotic aspects of the Teichmüller TQFT

We calculate the knot invariant coming from the Teichmüller TQFT [AK1]. Specifically we calculate the knot invariant for the complement of the knot $6_1$ both in the original [AK1] and the new formulation of the Teichmüller TQFT [AK2] for the one-vertex H-triangulation of $(S^3,6_1)$. We show that the two formulations give equivalent answers. Furthermore we apply a formal stationary phase analysis and arrive at the Andersen- Kashaev volume conjecture as stated in [AK1, Conj. 1]. Furthermore we calculate the first examples of knot complements in the new formulation showing that the new formulat…

Arcs on spheres intersecting at most twice

Let p be a puncture of a punctured sphere, and let Q be the set of all other punctures. We prove that the maximal cardinality of a set of arcs pairwise intersecting at most once, which start at p and end in Q, is |X|(|X| + 1). We deduce that the maximal cardinality of a set of arcs with arbitrary endpoints pairwise intersecting at most twice is |X|(|X| + 1)(|X| + 2).

穿孔球面上的收缩极值

Systoles on punctured spheres

本文确定了在赋予有限面积完备黎曼度量的$n$穿孔球面上，收缩的最大数量。

A discreteness algorithm for 4-punctured sphere groups

Let $Γ$ be a subgroup of $PSL(2,R)$ generated by three parabolic transformations. The main goal of this paper is to present an algorithm to determine whether or not $Γ$ is discrete. Historically discreteness algorithms have been considered within several broader mathematical paradigms: the discreteness problem, the construction and deformation of hyperbolic structures on surfaces and notions of automata for groups. Each of these approaches yield equivalent results. The second goal of this paper is to give an exposition of the basic ideas needed to interpret these equivalences, emphasizing rela…

焊接图、Wirtinger群与纽结穿孔球面

Welded graphs, Wirtinger groups and knotted punctured spheres

本文建立焊接图的图示理论并扩展 Satoh 的 Tube 映射，用以分类纽结穿孔球面、刻画其 Wirtinger 群，并提出保持 Tube 映射不变的 Υ 局部移动。

On the virtually-cyclic dimension of mapping class groups of punctured spheres

We calculate the virtually-cyclic dimension of the mapping class group of a sphere with at most six punctures. As an immediate consequence, we obtain the virtually-cyclic dimension of the mapping class group of the twice-holed torus and of the closed genus-two surface.

四穿孔球面的Bonahon-Wong-Yang体积猜想

The Bonahon-Wong-Yang volume conjecture for the four-puncture sphere

本文基于Checkhov-Fock代数表示理论显式计算了四穿孔球面自微分同胚的Bonahon-Wong-Yang量子不变量，并在特定技术条件下验证了四穿孔球面丛的体积猜想，提供了首批示例族证据。

Finite braid group orbits in Aff(C)-character varieties of the punctured sphere

We give a complete description of finite braid group orbits in Aff(C)-character varieties of the punctured Riemann sphere. This is performed thanks to a coalescence procedure and to the theory of finite complex reflection groups. We then derive consequences in the theory of differential equations. These concern algebraicity of isomonodromic deformations for reducible rank two logarithmic connections on the sphere, the Riemann-Hilbert problem and F_D-type Lauricella hypergeometric functions.

On the traceless SU(2) character variety of the 6-punctured 2-sphere

We exhibit the traceless $SU(2)$ character variety of a 6-punctured 2-sphere as a 2-fold branched cover of ${\mathbb{C}}P^3$, branched over the singular Kummer surface, with the branch locus in $R(S^2,6)$ corresponding to the binary dihedral representations. This follows from an analysis of the map induced on $SU(2)$ character varieties by the 2-fold branched cover $F_{n-1}\to S^2$ branched over $2n$ points, combined with the theorem of Narasimhan-Ramanan which identifies $R(F_2)$ with ${\mathbb{C}} P^3$. The singular points of $R(S^2,6)$ correspond to abelian representations, and we prove tha…

On the ideal triangulation graph of a punctured surface

We study the ideal triangulation graph $T(S)$ of a punctured surface $S$ of finite type. We show that if $S$ is not the sphere with at most three punctures or the torus with one puncture, then the natural map from the extended mapping class group of $S$ into the simplicial automorphism group of $T(S)$ is an isomorphism. We also show that under the same conditions on $S$, the graph $T(S)$ equipped with its natural simplicial metric is not Gromov hyperbolic. Thus, from the point of view of Gromov hyperbolicity, the situation of $T(S)$ is different from that of the curve complex of $S$.

Profinite刚性及圆周上的双曲四穿孔球丛

Profinite rigidity and hyperbolic four-punctured sphere bundles over the circle

本文证明了圆周上的双曲四穿孔球丛可由其基本群的有限商群在所有3-流形群中被唯一区分，并通过改进Liu的结果证明了纤维拓扑类型可由profinite完备性检测。

An extension of the Maskit slice for 4-dimensional Kleinian groups

Let $Γ$ be a 3-dimensional Kleinian punctured torus group with ccidental parabolic transformations. The deformation space of $Γ$ in the group of Möbius transformations on the 2-sphere is well-known as the Maskit slice of punctured torus groups. In this paper, we study deformations $Γ'$ of $Γ$ in the group of Möbius transformations on the 3-sphere such that $Γ'$ does not contain screw parabolic transformations. We will show that the space of the deformations is realized as a domain of 3-space $\mathbb{R}^3$, which contains the Maskit slice of punctured torus groups as a slice through a plane. F…

The normal closure of a power of a half-twist has infinite index in the mapping class group of a punctured sphere

In this paper we show that the normal closure of the mth power of a half-twist has infinite index in the mapping class group of a punctured sphere. Furthermore, in some cases we prove that the quotient of the mapping class group of the punctured sphere by the normal closure of a power of a half-twist contains a free abelian subgroup. As a corollary we prove that the quotient of the hyperelliptic mapping class group of a surface of genus at least two by the normal closure of the mth power of a Dehn twist has infinite order, and for some integers m the quotient contains a free nonabelian subgrou…

穿孔球面相对特征簇中的紧连通分支

Compact connected components in relative character varieties of punctured spheres

本研究证明穿孔球面基本群到$\text{SU}(p,q)$的相对特征簇存在紧连通分支，推广了此前$\text{PU}(1,1)$情形的结果，依托非阿贝尔Hodge对应证明并给出几何不变量理论下的显式构造。

Normal all pseudo-Anosov subgroups of mapping class groups

We construct the first known examples of nontrivial, normal, all pseudo-Anosov subgroups of mapping class groups of surfaces. Specifically, we construct such subgroups for the closed genus two surface and for the sphere with five or more punctures. Using the branched covering of the genus two surface over the sphere and results of Birman and Hilden, we prove that a reducible mapping class of the genus two surface projects to a reducible mapping class on the sphere with six punctures. The construction introduces "Brunnian" mapping classes of the sphere, which are analogous to Brunnian links.

Tykhyy关于有限映射类群轨道的猜想

Tykhyy's Conjecture on finite mapping class group orbits

本文通过归纳法证明了关于穿孔球面的映射类群作用在Deroin-Tholozan表示上的有限轨道分类，恢复了Tykhyy对5穿孔球面的分类，并完成了Tykhyy关于SL₂C表示的猜想证明。

The lower central and derived series of the braid groups of the sphere and the punctured sphere

Our aim is to determine the lower central series (LCS) and derived series (DS) for the braid groups of the sphere and of the finitely-punctured sphere. We show that for all n (resp. all n\geq 5), the LCS (resp. DS) of the n-string braid group B\_n(S^2) is constant from the commutator subgroup onwards, and that Γ\_2(B\_4(S^2)) is a semi-direct product of the quaternion group by a free group of rank 2. For n=4, we determine the DS of B\_4(S^2), as well as its quotients. For n \geq 1, the class of m-string braid groups B\_m(S^2) \ {x\_1,...,x\_n} of the n-punctured sphere includes the Artin braid…

On the arc and curve complex of a surface

We study the {\it arc and curve} complex $AC(S)$ of an oriented connected surface $S$ of finite type with punctures. We show that if the surface is not a sphere with one, two or three punctures nor a torus with one puncture, then the simplicial automorphism group of $AC(S)$ coincides with the natural image of the extended mapping class group of $S$ in that group. We also show that for any vertex of $AC(S)$, the combinatorial structure of the link of that vertex characterizes the type of a curve or of an arc in $S$ that represents that vertex. We also give a proof of the fact if $S$ is not a sp…

随机穿孔双曲面与布朗球面

Random punctured hyperbolic surfaces & the Brownian sphere

研究按Weil-Petersson测度采样的n+1个穿孔0类随机双曲面Sₙ，证明度量经n⁻¹/⁴缩放后收敛于布朗球面，未缩放时局部收敛于特定无限体积双曲面，通过平面树编码等技术实现。

关于带孔曲面的 $SU(2)$ 特征簇的辛方面

On symplectic aspects of $SU(2)$ character varieties for punctured surfaces

本文研究带奇数孔曲面的 $SU(2)$ 平坦联络模空间的辛性质，证明映射类群到辛映射类群的同态在维数大于2时是单射，并给出5点爆破射影平面中拉格朗日球面的完全分类及其可移动性判定。

The quandle of the trefoil knot as the Dehn quandle of the torus

We prove that the fundamental quandle of the trefoil knot is isomorphic to the projective primitive subquandle of transvections of the symplectic space $\Z \oplus \Z$. The last quandle can be identified with the Dehn quandle of the torus and the cord quandle on a 2-sphere with four punctures. We also show that the fundamental quandle of the long trefoil knot is isomorphic to the cord quandle on a 2-sphere with a hole and three punctures.

Non-unitary TQFTs from 3D $\mathcal{N}=4$ rank 0 SCFTs

We propose a novel procedure of assigning a pair of non-unitary topological quantum field theories (TQFTs), TFT$_\pm [\mathcal{T}_{\rm rank \;0}]$, to a (2+1)D interacting $\mathcal{N}=4$ superconformal field theory (SCFT) $\mathcal{T}_{\rm rank \;0}$ of rank 0, i.e. having no Coulomb and Higgs branches. The topological theories arise from particular degenerate limits of the SCFT. Modular data of the non-unitary TQFTs are extracted from the supersymmetric partition functions in the degenerate limits. As a non-trivial dictionary, we propose that $F = \max_α\left(- \log |S^{(+)}_{0α}| \right) = …

Injections of Artin groups

We study those Artin groups which, modulo their centers, are finite index subgroups of the mapping class group of a sphere with at least 5 punctures. In particular, we show that any injective homomorphism between these groups is parameterized by a homeomorphism of a punctured sphere together with a map to the integers. We also give a generating set for the automorphism group of the pure braid group on at least 4 strands. The technique, following Ivanov, is to prove that every superinjective map of the complex of curves of a sphere with at least 5 punctures is induced by a homeomorphism.

2-systems of arcs on spheres with prescribed endpoints

Let $S$ be an $n$-punctured sphere, with $n \geq 3$. We prove that $\binom{n}{3}$ is the maximum size of a family of pairwise non-homotopic simple arcs on $S$ joining a fixed pair of distinct punctures of $S$ and pairwise intersecting at most twice. On the way, we show that a square annular diagram $A$ has a corner on each of its boundary paths if $A$ contains at least one square and the dual curves of $A$ are simple arcs joining the boundary paths of $A$ and pairwise intersecting at most once.

An analytic family of representations for the mapping class group of punctured surfaces

We use quantum invariants to define an analytic family of representations for the mapping class group of a punctured surface. The representations depend on a complex number A with |A| <= 1 and act on an infinite-dimensional Hilbert space. They are unitary when A is real or imaginary, bounded when |A|<1, and only densely defined when |A| = 1 and A is not a root of unity. When A is a root of unity distinct from 1, -1, i, -i the representations are finite-dimensional and isomorphic to the "Hom" version of the well-known TQFT quantum representations. The unitary representations in the interval [-1…

Line and surface defects in Reshetikhin-Turaev TQFT

A modular tensor category $\mathcal{C}$ gives rise to a Reshetikhin-Turaev type topological quantum field theory which is defined on 3-dimensional bordisms with embedded $\mathcal{C}$-coloured ribbon graphs. We extend this construction to include bordisms with surface defects which in turn can meet along line defects. The surface defects are labelled by $Δ$-separable symmetric Frobenius algebras and the line defects by "multi-modules" which are equivariant with respect to a cyclic group action. Our invariant cannot distinguish non-isotopic embeddings of 2-spheres, but we give an example where …

小曲面 skein 代数的显式表示与 Azumaya 轨迹

Explicit representations and Azumaya loci of skein algebras of small surfaces

本文通过显式公式构造单孔环面与四孔球面 skein 代数在所有单位根处的有限维表示，并利用其可约性确定 Azumaya 轨迹，揭示了其与经典阴影簇光滑轨迹的包含关系。

Representations of Motion Groups of Links via Dimension Reduction of TQFTs

Motion groups of links in the three sphere $\mathbb{S}^3$ are generalizations of the braid groups, which are motion groups of points in the disk $\mathbb{D}^2$. Representations of motion groups can be used to model statistics of extended objects such as closed strings in physics. Each $1$-extended $(3+1)$-topological quantum field theory (TQFT) will provide representations of motion groups, but it is difficult to compute such representations explicitly in general. In this paper, we compute representations of the motion groups of links in $\mathbb{S}^3$ with generalized axes from Dijkgraaf-Witt…

Monodromy groups of $\mathbb{C}\mathrm{P}^1$-structures on punctured surfaces

For a punctured surface $S$, we characterize the representations of its fundamental group into $\mathrm{PSL}_2 (\mathbb{C})$ that arise as the monodromy of a meromorphic projective structure on $S$ with poles of order at most two and no apparent singularities. This proves the analogue of a theorem of Gallo-Kapovich-Marden concerning $\mathbb{C}\mathrm{P}^1$-structures on closed surfaces, and settles a long-standing question about characterizing monodromy groups for the Schwarzian equation on punctured spheres. The proof involves a geometric interpretation of the Fock-Goncharov coordinates of t…

流形的SKK群与非酉可逆拓扑量子场论

SKK groups of manifolds and non-unitary invertible TQFTs

本文通过研究带切向结构流形的SKK群，解决了奇欧拉示性数流形存在性与短正合序列分裂性两个关键问题，并据此完成十重分类中1-5维非酉可逆拓扑量子场论的完整分类。

映射类群模扭转幂的刚性

Rigidity of mapping class groups mod powers of twists

本文研究带孔球面映射类群模Dehn扭转幂的商，证明其自同构与拟等距刚性，并发展分层双曲空间技术以给出裤子图刚性的新证明。

Super-maximal representations from fundamental groups of punctured surfaces to $\mathrm{PSL}(2,\mathbb{R})$

We study a particular class of representations from the fundamental groups of punctured spheres $Σ_{0,n}$ to the group $\text{PSL} (2,\mathbb R)$ (and their moduli spaces), that we call \emph{super-maximal}. Super-maximal representations are shown to be \emph{totally non hyperbolic}, in the sense that every simple closed curve is mapped to a non hyperbolic element. They are also shown to be \emph{geometrizable} (appart from the reducible super-maximal ones) in the following very strong sense : for any element of the Teichmüller space $\mathcal T_{0,n}$, there is a unique holomorphic equivarian…