arXiv research

Algorithm decides if genus-two surfaces in 3-sphere are isotopic.

We show that the only closed 4-manifolds admitting genus two trisections are $S^2 \times S^2$ and connected sums of $S^1 \times S^3$, $\mathbb{CP}^2$, and $\overline{\mathbb{CP}}^2$ with two summands. Moreover, each of these manifolds admits a unique genus two trisection up to diffeomorphism. The proof relies heavily o…

Classifies Teichmüller curves in genus three from genus two.

We compute the class of arithmetic genus two Teichmueller curves in the Picard group of pseudo-Hilbert modular surfaces, distinguished according to their torsion order and spin invariant. As an application, we compute the number of genus two square-tiled surfaces with these invariants. The main technical tool is the co…

We show that all GL(2, R)-equivariant point markings over orbit closures of primitive genus two translation surfaces arise from marking pairs of points exchanged by the hyperelliptic involution, Weierstrass points, or the golden points in the golden eigenform locus. As corollaries, we classify the holomorphically varyi…

Extremal length systole is maximized at the Bolza surface.

It is known that there are surface bundles of arbitrarily high genus which have genus two Heegaard splittings. The simplest examples are Seifert fibered spaces with the sphere as a base space, three exceptional fibers and which allow horizontal surfaces. We characterize the monodromy maps of all surface bundles with ge…

We provide a classification of the essential surfaces of non-negative Euler characteristic in the exteriors of genus two handlebodies embedded in the 3-sphere.

We study the existence of incompressible embeddings of surfaces into the genus two handlebody. We show that for every compact surface with boundary, orientable or not, there is an incompressible embedding of the surface into the genus two handlebody. In the orientable case the embedding can be either separating or non-…

We give a construction of hyperbolic 3-manifolds with rank two fundamental groups and report an experimental search to find such manifolds. Our manifolds are all surface bundles over the circle with genus two surface fiber. For the manifolds so obtained, we then examine whether they are of Heegaard genus two or not. As…

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 an…

A proof that the separating curve complex of the closed genus two surface has a quasi-distance formula and is delta hyperbolic using tools of Masur and Schleimer. This answers in the affirmative a Conjecture of Schleimer.

In this paper, we characterize closed incompressible surfaces of genus two in the complements of 3-bridge knots and links. This characterization includes that of essential 2-string tangle decompositions for 3-bridge knots and links.

In this note we find new relations in the mapping class group of a genus two surface with n boundary components for n=1,..., 8 which induce a genus two Lefschetz fibration $CP^2#13CP^2bar \to S^2$ with n disjoint sections. As a consequence, we observe any holomorphic genus 2 Lefschetz fibration without separating singu…

The paper studies Goeritz equivalence in lens spaces, describing actions and obstructions.

This paper classifies and determines the length of the shortest filling pairs on a specific type of surface.

We give a necessary and sufficient condition for a simple closed curve on the boundary of a genus two handlebody to decompose the handlebody into (torus with one boundary component times [0,1]. We use this condition to decide whether a simple closed curve on a genus two Heegaard surface is a GOF-knot (genus one fibered…

Let S be a connected, compact and orientable surface of genus two having exactly one boundary component. We study automorphisms of the Torelli complex for S, and describe any isomorphism between finite index subgroups of the Torelli group for S. More generally, we study superinjective maps from the Torelli complex for …

We calculate the Euler characteristics of all of the Teichmuller curves in the moduli space of genus two Riemann surfaces which are generated by holomorphic one-forms with a single double zero. These curves can all be embedded in Hilbert modular surfaces and our main result is that the Euler characteristic of a Teichmu…

In this paper we construct a faithful representation of the mapping class group of the genus two surface into a group of matrices over the complex numbers. Our starting point is the Lawrence-Krammer representation of the braid group B_n, which was shown to be faithful by Bigelow and Krammer. We obtain a faithful repres…

Defines pants distance for knotted surfaces in 4-manifolds.

Paper shows Seiberg-Witten invariants vanish for Davis hyperbolic 4-manifold.

We generalize McShane's identity for the length series of simple closed geodesics on a cusped hyperbolic surface to hyperbolic cone-surfaces (with all cone angles $\le π$), possibly with cusps and/or geodesic boundary. In particular, by applying the generalized identity to the orbifolds obtained from taking the quotien…

In this paper, we describe the relation between the study of closed connected surfaces embedded in $S^3$ and the theory of handlebody-knots. By Fox's theorem, a pair of handlebody-knots is associated to a closed connected surface embedded in $S^3$ in the sense that their exterior components are pairwise homeomorphic. W…

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…

In this article, we provide bounds on systoles associated to a holomorphic $1$-form $ω$ on a Riemann surface $X$. In particular, we show that if $X$ has genus two, then, up to homotopy, there are at most $10$ systolic loops on $(X,ω)$ and, moreover, that this bound is realized by a unique translation surface up to homo…

In 1996 M. Traizet obtained singly periodic minimal surfaces with Scherk ends of arbitrary genus by desingularizing a set of vertical planes at their intersections. However, in Traizet's work it is not allowed that three or more planes intersect at the same line. In our paper, by a {\it saddle-tower} we call the desing…

Proof that $SU(2)$ character variety of genus 2 surface is ${\mathbb C} P^3$.

Suppose that S is a surface of genus two or more, with exactly one boundary component. Then the curve complex of S has one end.

Let M be a closed, irreducible, genus two 3-manifold, and F a maximal collection of pairwise disjoint, closed, orientable, incompressible surfaces embedded in M. Then each component manifold M_i of M-F has handle number at most one, i.e. admits a Heegaard splitting obtained by attaching a single 1-handle to one or two …

The goal of these notes is to prove that the mapping class groups of a closed orientable surface of genus two, with punctures, are not Kähler

Square-tiled surfaces can be classified by their number of squares and their cylinder diagrams (also called realizable separatrix diagrams). For the case of $n$ squares and two cone points with angle $4 π$ each, we set up and parametrize the classification into four diagrams. Our main result is to provide formulae for …

We study the congruence problem for subgroups of the modular group that appear as Veech groups of square-tiled surfaces in the minimal stratum of abelian differentials of genus two.

Every fibration of a projective hyper-Kähler fourfold has fibers which are Abelian surfaces. In case the Abelian surface is a Jacobian of a genus two curve, these have been classified by Markushevich. We study those cases where the Abelian surface is a product of two elliptic curves, under some mild genericity hypothes…

In this paper, we settle in the affirmative the Jakobson-Levitin-Nadirashvili-Nigam-Polterovich conjecture, stating that a certain singular metric on the Bolza surface, with area normalized, should maximize the first eigenvalue of the Laplacian.

In this short note, we construct a minimally intersecting pair of simple closed curves that fill a genus 2 surface with an odd, greater than 3, number of punctures. This finishes the determination of minimally intersecting filling pairs for all surfaces completing the work of Aougab-Huang and Aougab-Taylor.

Correspondence found between Askey-Wilson polynomials and genus-two handlebody skein module.

A Teichmüller curve is an algebraic and isometric immersion of an algebraic curve into the moduli space of Riemann surfaces. We give the first explicit algebraic models of Teichmüller curves of positive genus. Our methods are based on the study of certain Hilbert modular forms and the use of Ahlfors's variational formu…

Study shows genus two Heegaard diagrams for all Thurston geometries.

Enumerated all genus two handlebody-knots with seven crossings.

We apply topological methods to study the smallest non-zero number $λ_1$ in the spectrum of the Laplacian on finite area hyperbolic surfaces. For closed hyperbolic surfaces of genus two we show that the set $\{S \in {\mathcal{M}_2}: {λ_1}(S) > 1/4 \}$ is unbounded and disconnects the moduli space ${\mathcal{M}_2}$.

One can embed arbitrarily many disjoint, non-parallel, non-boundary parallel, incompressible surfaces in any three manifold with at least one boundary component of genus two or greater [4]. This paper proves the contrasting, but not contradictory result that although one can sometimes embed arbitrarily many surfaces in…

We describe an example of a closed orientable 3-manifold with distinct distance three genus two Heegaard splittings. This demonstrates that the constructions of alternate genus two Heegaard splittings of closed orientable 3-manifolds described by Rubinstein and Scharlemann in their 1998 paper Genus Two Heegaard Splitti…

We compute the genus one family Gromov-Witten invariants of K3 surfaces for non-primitive classes. These calculations verify Gottsche-Yau-Zaslow formula for non-primitive classes with index two. Our approach is to use the genus two topological recursion formula and the symplectic sum formula to establish relationships …

We find all Heegaard diagrams with the property "alternating" or "weakly alternating" on a genus two orientable closed surface. Using these diagrams we give infinitely many genus two 3--manifolds, each admits an automorphism whose non-wondering set consists of two Williams solenoids, one attractor and one repeller. The…

We prove formulae for the countings by orbit of square-tiled surfaces of genus two with one singularity. These formulae were conjectured by Hubert & Lelièvre. We show that these countings admit quasimodular forms as generating functions.

We study Veech surfaces of genus 2 arising from quadratic differentials that are not squares of abelian differentials. We prove that all such surfaces of type (2,2) and (2,1,1) are arithmetic. In (1,1,1,1) case, we reduce the question to abelian differentials of type (2,2) on hyperelliptic genus 3 surfaces with singula…

We study the Calabi functional on a ruled surface over a genus two curve. For polarisations which do not admit an extremal metric we describe the behaviour of a minimising sequence splitting the manifold into pieces. We also show that the Calabi flow starting from a metric with suitable symmetry gives such a minimising…