arXiv research

Researchers compute TQFT representation for sphere with 4 punctures.

Classifies arcs on a 4-punctured sphere that intersect at most once.

The paper presents an algorithm to determine discreteness of certain groups.

The paper constructs solutions for Higgs fields on a 4-punctured sphere.

Researchers prove positivity of skein algebra structure constants for specific surfaces.

Classifies finite orbits of mapping class group action on character varieties.

An earlier article with Francis Bonahon introduced new invariants for pseudo-Anosov diffeomorphisms of surface, based on the representation theory of the quantum Teichmuller space. We explicity compute these quantum hyperbolic invariants in the case of the 1-puncture torus and the 4-puncture sphere.

Following Riley's work, for each 2-bridge link $K(r)$ of slope $r\in\QQ$ and an integer or a half-integer $n$ greater than 1, we introduce the {\it Heckoid orbifold $\orbs(r;n)$} and the {\it Heckoid group $\Hecke(r;n)=π_1(\orbs(r;n))$ of index $n$ for $K(r)$}. When $n$ is an integer, $\orbs(r;n)$ is called an {\it eve…

With a 4-ended tangle $T$, we associate a Heegaard Floer invariant $\operatorname{CFT^\partial}(T)$, the peculiar module of $T$. Based on Zarev's bordered sutured Heegaard Floer theory, we prove a glueing formula for this invariant which recovers link Floer homology $\operatorname{\widehat{HFL}}$. Moreover, we classify…

Given a pointed 4-ended tangle $T \subset D^3$, there are two Khovanov theoretic tangle invariants, $\unicode{1044}_1(T)$ from [arXiv:1910.1458] and $L_T$ from [arXiv:1808.06957], which are twisted complexes over the Fukaya category of the boundary 4-punctured sphere $(S^2,4\text{pt})=\partial (D^3, T)$. We prove that …

The paper provides presentations for mapping class groups and cluster automorphism groups of surfaces.

For $n$ an even number, we study representations of the mapping class group of the $n$-punctured sphere arising from $\mathrm{SU}(2)$-TQFT when all punctures are colored by the same integer $N \geq 1$. We prove that the conjecture of Andersen, Masbaum and Ueno holds for the $4$-punctured sphere for all $N \geq 2$. In t…

We use Bonahon-Wong's trace map to study character varieties of the once-punctured torus and of the 4-punctured sphere. We clarify a relationship with cluster algebra associated with ideal triangulations of surfaces, and we show that the Goldman Poisson algebra of loops on surfaces is recovered from the Poisson structu…

We investigate the Lawson genus $2$ surface by methods from integrable system theory. We prove that the associated family of flat connections comes from a family of flat connections on a $4-$punctured sphere. We describe the symmetries of the holonomy and show that it is already determined by the holonomy around one of…

The paper connects hyperpolygon spaces to Higgs bundle moduli spaces via degenerations.

Operator on tangles derived from knot 2-cabling.

Quantum theory of curved tetrahedrons yields quantum group intertwiners.

This paper gives the first explicit, two-sided estimates on the cusp area of once-punctured torus bundles, 4-punctured sphere bundles, and 2-bridge link complements. The input for these estimates is purely combinatorial data coming from the Farey tesselation of the hyperbolic plane. The bounds on cusp area lead to expl…

We study a topological aspect of rank-1 double affine Hecke algebra (DAHA). Clarified is a relationship between the DAHA of A1-type (resp. CC1-type) and the skein algebra on a once-punctured torus (resp. a 4-punctured sphere), and the SL(2;Z) actions of DAHAs are identified with the Dehn twists on the surfaces. Combini…

Given a 2-stranded tangle in a $\ZZ/2$ homology ball, $T\subset Y$, we investigate the character variety $R(Y,T)$ of conjugacy classes of traceless SU(2) representations of $π_1(Y\setminus T)$. In particular we completely determine the subspace of binary dihedral representations, and identify all of $R(Y,T)$ for many t…

Analyzes convex structures in Teichmüller space unit tangent spheres.

Paper constructs Lawson surfaces using Fuchsian DPW potentials.

The purpose of this thesis is to define a "local" version of Ozsváth and Szabó's Heegaard Floer homology $\operatorname{\widehat{HFL}}$ for links in the 3-dimensional sphere, i.e. a Heegaard Floer homology $\operatorname{\widehat{HFT}}$ for tangles in the closed 3-ball. After studying basic properties of $\operatorname…

Geometric interpretation of tangle invariants using immersed curves.

New actions found on exotic spheres using group theory.

For every N > 0 there exists a group of deficiency less than -N that arises as the fundamental group of a smooth homology 4-sphere and also as the fundamental group of the complement of a compact contractible submanifold of the 4-sphere. A group is the fundamental group of the complement of a contractible submanifold o…

We show that any simply connected topological closed $4$-manifold punctured along any compact, totally disconnected tame subset $Λ$ admits a continuum of smoothings which are not diffeomorphic to any leaf of a $C^{1,0}$ codimension one foliation on a compact manifold. This includes the remarkable case of $S^4$ puncture…

Given a genus-g Heegaard splitting of a 3-sphere, the genus-g Goeritz group is defined to be the group of the isotopy classes of orientation preserving homeomorphism of the 3-sphere that preserve the splitting. In this paper, we determine the twisted first (co)homology group of the genus-2 Goeritz group of 3-sphere.

The only finite nonabelian simple group acting on a homology 3-sphere - necessarily non-freely - is the dodecahedral group $\Bbb A_5 \cong {\rm PSL}(2,5)$ (in analogy, the only finite perfect group acting freely on a homology 3-sphere is the binary dodecahedral group $\Bbb A_5^* \cong {\rm SL}(2,5)$). In the present pa…

Classifies non-linear Fredholm maps linking to stable homotopy groups of spheres.

A finite nonabelian simple group does not admit a free action on a homology sphere, and the only finite simple group which acts on a homology sphere with at most 0-dimensional fixed point sets ("pseudofree action") is the alternating group A_5 acting on the 2-sphere. Our first main theorem is the finiteness result that…

Formula derived for discrete improper affine spheres.

Proves constraints on groups extending Möbius transformations on spheres.

Study on group cocycles for volume-preserving diffeomorphisms.

Study Goeritz groups of link decompositions, focusing on their asymptotic behavior.

It is a consequence of the classical Jordan bound for finite subgroups of linear groups that in each dimension n there are only finitely many finite simple groups which admit a faithful, linear action on the n-sphere. In the present paper we prove an analogue for smooth actions on arbitrary homology n-spheres: in each …

New rack and multiple group rack cohomology for surfaces in 3-sphere.

Hyperbolic groups act on spheres, and nearby actions are semi-conjugate.

Contact group retracts to unitary subgroup.

We establish an interesting connection between Morin singularities and stable homotopy groups of spheres. We apply this connection to computations of cobordism groups of certain singular maps. The differentials of the spectral sequence computing these cobordism groups are given by the composition multiplication in the …

We prove that all smooth sphere bundles that admit fiberwise 1/4-pinched metrics are induced bundles of vector bundles, so their structure groups reduce from the diffeomorphism group of the sphere to the orthogonal group. This result implies the existence of many smooth n-sphere bundles over a k-sphere that do not supp…

Study automorphisms of pure braid groups on sphere homotopy groups.

Study extends symmetries of sphere points to surface mapping classes.

Study invariants of $\mathbb{Z}/p$-homology 3-spheres from abelianization of mapping class groups.

The paper proves group actions on spheres with odd fixed points.

In Heisenberg group, bisectors are spinal spheres with specific curvature.

The paper connects diffeomorphism groups and sphere embeddings, proving a group structure.

Authors prove a conjecture about the Goeritz group of 3-sphere Heegaard splittings.