arXiv research

This paper refines previous work by the first author. We study the question of which links in the 3-sphere can be obtained as closures of a given 1-manifold in an unknotted solid torus in the 3-sphere (or genus-1 tangle) by adjoining another 1-manifold in the complementary solid torus. We distinguish between even and o…

Study confirms non-injective monodromy for even genus 4 translation surfaces.

New 3D handlebodies in 4-sphere and 5-ball are not isotopic even with same boundary.

Study eigenvalues of curvature operators to annihilate cobordism invariants.

New Witten rigidity theorems for elliptic genus in various dimensions.

Researchers confirm a conjecture about metrics on a specific Teichmüller space.

New bounds on nonorientable four-ball genus for torus knots.

Involutions without fixed points on hyperbolic closed Riemann surface are discussed. For an orientable surface $X$ of even genus with an arbitrary Riemannian metric $d$ admitting an involution $τ$, it is known that $\min_{p\in X}d(p,τ(p))$ is bounded by a constant which depends on the genus of $X$. The equivalent resul…

We construct noncomplex smooth 4-manifolds which admit genus-2 Lefschetz fibrations over S^2. The fibrations are necessarily hyperelliptic, and the resulting 4-manifolds are not even homotopy equivalent to complex surfaces. Furthermore, these examples show that fiber sums of holomorphic Lefschetz fibrations do not nece…

New rules for classifying certain square roots of surface automorphisms.

There is a relation between the generalized Property R Conjecture and the Schoenflies Conjecture that suggests a new line of attack on the latter. The approach gives a quick proof of the genus 2 Schoenflies Conjecture and suffices to prove the genus 3 case, even in the absence of new progress on the generalized Propert…

New minimal surfaces found in 3D sphere with low genus.

We give an exponential upper and a quadratic lower bound on the number of pairwise non-isotopic simple closed curves can be placed on a closed surface of genus g such that any two of the curves intersects at most once. Although the gap is large, both bounds are the best known for large genus. In genus one and two, we s…

Triple-crossing number bound for knots and links, especially torus knots.

New examples show inequalities can be sharp even when self-linking and genus differ.

We compare the values of the nonorientable three genus (or, crosscap number) and the nonorientable four genus of torus knots. In particular, let T(p,q) be any torus knot with p even and q odd. The difference between these two invariants on T(p,q) is at least k/2, where p = qk + a and 0 < a < q and $k\geq 0$. Hence, the…

An important difference between high dimensional smooth manifolds and smooth 4-manifolds that in a 4-manifold it is not always possible to represent every middle dimensional homology class with a smoothly embedded sphere. This is true even among the simplest 4-manifolds: $X_0(K)$ obtained by attaching an $0$-framed 2-h…

Study bounds and asymptotic behavior of largest purely non-free actions on compact Riemann surfaces.

We show that any Kahler extension of a finitely generated abelian group by a surface group of genus g at least 2 is virtually a product. Conversely, we prove that any homomorphism of an even rank, finitely generated abelian group into the genus g mapping class group with finite image gives rise to a Kahler extension. T…

Lower bounds on average genus of 2-bridge knots found.

We prove that the sequence of projective representations of the mapping class group obtained from the projective flat connection in the SU(n)-Verlinde bundles over Teichmuller space is asymptotically faithful, that is the intersection over all levels of the kernels of these representations is trivial, whenever the genu…

The paper sets genus bounds for twisted quantum invariants.

New rigidity theorems for spin^c manifolds using modular invariance.

We prove the filling area conjecture in the hyperelliptic case. In particular, we establish the conjecture for all genus 1 fillings of the circle, extending P. Pu's result in genus 0. We translate the problem into a question about closed ovalless real surfaces. The conjecture then results from a combination of two ingr…

The study finds infinite knot exteriors with meridional surfaces of any genus and boundary components.

Suppose C is a singular curve in CP^2 and it is topologically an embedded surface of genus g; such curves are called cuspidal. The singularities of C are cones on knots K_i. We apply Heegaard Floer theory to find new constraints on the sets of knots {K_i} that can arise as the links of singularities of cuspidal curves.…

The paper shows how to generate mapping class groups with specific involutions.

We map out the moduli space of Lawson symmetric constant mean curvature surfaces in the 3-sphere of genus $g>1$ by flowing numerically from Delaunay tori with even lobe count via the generalized Whitham flow.

We show that the topological charge of the n-soliton solution of the sine-Gordon equation n is related to the genus g > 1 of a constant negative curvature compact surface described by this configuration. The relation is n=2(g-1), where n is even. The moduli space of complex dimension B(g)=3(g-1) corresponds precisely t…

The main goal of this article is to understand how the length spectrum of a random surface depends on its genus. Here a random surface means a surface obtained by randomly gluing together an even number of triangles carrying a fixed metric. Given suitable restrictions on the genus of the surface, we consider the number…

This note presents a formula for the enumerative invariants of arbitrary genus in toric surfaces. The formula computes the number of curves of a given genus through a collection of generic points in the surface. The answer is given in terms of certain lattice paths in the relevant Newton polygon. If the toric surface i…

We showed that the twist subgroup of the mapping class group of a closed connected nonorientable surface of genus $g\geq13$ can be generated by two involutions and an element of order $g$ or $g-1$ depending on whether $g$ is odd or even respectively.

This paper shows how to construct Abelian differentials with any prescribed singularities.

Equations of dispersionless Hirota type have been thoroughly investigated in the mathematical physics and differential geometry literature. It is known that the parameter space of integrable Hirota type equations in 3D is 21-dimensional and the action of the natural equivalence group Sp(6, R) on the parameter space has…

The paper proves the existence of maxfaces with multiple swallowtails and planar ends.

A genus-1 tangle G is an arc properly embedded in a standardly embedded solid torus S in the 3-sphere. We say that a genus-1 tangle embeds in a knot K in S^3 if the tangle can be completed by adding an arc exterior to the solid torus to form the knot K. We call K a closure of G. An obstruction to embedding a genus-1 ta…

For a closed 4-manifold $X$ and a knot $K$ in the boundary of punctured $X$, we define $γ_X^0(K)$ to be the smallest first Betti number of non-orientable and null-homologous surfaces in punctured $X$ with boundary $K$. Note that $γ^0_{S^4}$ is equal to the non-orientable 4-ball genus and hence $γ^0_X$ is a generalizati…

The paper proves conditions for minimal surfaces to be holomorphic and stable.

A manifold which admits a reducible genus-$2$ Heegaard splitting is one of the $3$-sphere, $S^2 \times S^1$, lens spaces or their connected sums. For each of those splittings, the complex of Haken spheres is defined. When the manifold is the $3$-sphere, $S^2 \times S^1$ or the connected sum whose summands are lens spac…

This thesis proves how to generate a specific group using involutions.

The study finds all trace field degrees for Torelli group mappings.

Starting from works by Scherk (1835) and by Enneper-Weierstraß\ (1863), new minimal surfaces with Scherk ends were found only in 1988 by Karcher (see \cite{Karcher1,Karcher}). In the singly periodic case, Karcher's examples of positive genera had been unique until Traizet obtained new ones in 1996 (see \cite{Traizet}).…

Disk and sphere graphs embed quasi-isometrically in R^2.

Study essential surfaces in knot exteriors, proving some combinations are realizable.

Let G be a finite group acting orthogonally on a pair (S^d,Γ) where Γis a finite, connected graph of genus g>1 embedded in the sphere S^d. The 3-dimensional case d=3 has recently been considered in a paper by C. Wang, S. Wang, Y. Zhang and the present author where for each genus g>1 the maximum order of a G-action on a…

This note gives a brief survey of the minimum dilatation problem for pseudo-Anosov mapping classes, and the first explicit train track description of an infinite family of pseudo-Anosov mapping classes with orientable stable foliations and the conjectural minimum dilatation for closed surfaces of even genus $g \ge 2$.

Maximal dilatation found on nonorientable surfaces.

Let Ng be the connected closed nonorientable surface of genus g >= 5 and Mod(Ng) denote the mapping class group of Ng. We prove that the outer automorphism group of Mod(Ng) is either trivial or Z if g is odd, and injects into the mapping class group of sphere with four holes if g is even.