arXiv research

Characterizes compact complex surfaces with finite homotopy rank-sum.

Study Murasugi sum in 4D for knotted surfaces, defining arborescent surfaces.

Study end sum for surfaces and prove uniqueness results.

Defines state sum models with defects in 3-manifolds.

Paper calculates L-invariant and L*-invariant for complex surface sums.

Characterizes Stein surfaces with finite homotopy rank-sum.

For each pair of integers g at least 2 and h at least 1, we explicitly construct infinitely many fiber sum and section sum indecomposable genus g surface bundles over genus h surfaces whose total spaces are pairwise homotopy inequivalent.

For each g > 2 and h > 1, we explicitly construct (1) fiber sum indecomposable relatively minimal genus g Lefschetz fibrations over genus h surfaces whose monodromies lie in the Torelli group, (2) fiber sum indecomposable genus g surface bundles over genus h surfaces whose monodromies are in the Torelli group (provided…

Quantum modularity proved for SU(2) TQFT signature on genus 2 surfaces.

Solves Demailly's system for direct sums of ample line bundles on Riemann surfaces.

The height function of various surfaces decomposes into finite sums of scaled and translated versions of itself.

The paper defines conditions for surface sums of handlebodies to be handlebodies.

The formula for the Turaev-Viro invariant of a 3-manifold depends on a complex parameter t. When t is not a root of unity, the formula becomes an infinite sum. This paper analyzes convergence of this sum when t does not lie on the unit circle, in the presence of an efficient triangulation of the three-manifold. The ter…

Knots can be constructed and decomposed using Murasugi sums of Seifert surfaces.

The study classifies strongly irreducible Heegaard splittings in hyperbolic 3-manifolds.

Let $S_g$ be a closed surface of genus $g$ and let $(α, β)$ be a filling pair on $S_g$; then $i(α, β) \geq 2g-1$, where $i$ is the (geometric) intersection number. Aougab and Huang demonstrated that (exponentially many) minimally-intersecting filling pairs exist on $S_g$ when $g > 2$ by a construction w…

New cosmological spacetimes without CMC Cauchy surfaces found.

Example of non-smooth isotopy using K3 surfaces.

Let M denote the total space of a Lefschetz fibration, obtained by blowing up a Lefschetz pencil on an algebraic surface. We consider the n-fold fibre sum M(n), generalizing the construction of the elliptic surfaces E(n). For a Lefschetz pencil on a simply-connected minimal surface of general type we partially calculat…

In this paper we determine the integral homology and cohomology groups of a closed 4-manifold X obtained as the generalized fibre sum of two closed 4-manifolds M and N along embedded surfaces of genus g and self-intersection zero. If the homologies of the 4-manifolds are torsion free and the surfaces represent indivisi…

Study on descent properties of complex affine surfaces under proper morphisms.

In this paper we define a new state sum based on the regions defined by tangles on a surface which is an oriented closed surface with a finite number of open holes drilled. From this state sum we obtain an invariant of regular isotopy for the tangles named $u$-invariant. The values of the $u$-invariant are in $\mathbb{…

Any two homologous surfaces of the same genus embedded in a smooth 4-manifold X with simply-connected complements are shown to be smoothly isotopic in the connected sum of X and the product of a 2-sphere with itself, if the surfaces are ordinary, and in the connected sum of X with the non-trivial sphere bundle over the…

Paper bounds the sum of index and nullity of minimal surfaces in certain 3-manifolds.

The 2-twist spun trefoil is an example of a sphere that is knotted in 4-dimensional space. Here this example is shown to be distinct from the same sphere with the reversed orientation. To demonstrate this fact a state-sum invariant for classical knots and knotted surfaces is developed via a cohomology theory of racks a…

The study sets lower bounds for eigenvalue sums of Laplacian on bounded domains and spheres.

Enhanced Khovanov TQFT using basepoints for nonorientable surfaces.

In this note we apply a 4-fold sum operation to develop an associativity rule for the pairwise symplectic sum. This allows us to show that certain diffeomorphic symplectic $4$-manifolds made out of elliptic surfaces are in fact symplectically deformation equivalent. We also show that blow-up points can be traded from o…

Classifies $T^2$-bundles over closed surfaces up to isomorphisms.

The paper defines new polynomials for links and linkoids.

Unified invariant for immersed surface-links using biquandle cocycles.

The paper introduces surface-complexity to measure 3-manifold complexity.

Classifies representations up to dimension 3g-3 for surface mapping class groups.

We establish a general `gluing theorem', which states roughly that if two nondegenerate constant mean curvature surfaces are juxtaposed, so that their tangent planes are parallel and very close to one another, but oppositely oriented, then there is a new constant mean curvature surface quite near to this configuration …

Study of convergence of point-object configurations to a charged dust continuum.

We prove that the only surfaces in $3$-dimensional Euclidean space $\R^3$ with constant Gaussian curvature $K$ and constructed by the sum of two space curves are cylindrical surfaces, in particular, $K=0$.

A knot is an a-small knot if its exterior does not contain closed incompressible surfaces disjoint from some incompressible Seifert surface for the knot. Using circular thin position for knots we prove that the handle number is additive under the connected sum of two a-small knots. As a consequence the Morse-Novikov nu…

Modulo trivial exceptions, we show that smoothly nontrivial symplectic sums of symplectic 4-manifolds along surfaces of positive genus are never rational or ruled, and we enumerate each case in which they have Kodaira dimension zero (i.e., are blowups of symplectic 4-manifolds with torsion canonical class). In particul…

Extends Rasmussen's invariant to new surfaces in four-manifolds.

We give an identity involving sums of functions of lengths of simple closed geodesics, known as a McShane identity, on any non-orientable hyperbolic surface with boundary which generalises Mirzakhani's identities on orientable hyperbolic surfaces with boundary.

Develops Brunn-Minkowski theory in hyperbolic space using hyperbolic p-sum.

We prove the Chern-Weil formula for SU(n+1)-singular connections over the complement of an embedded oriented surface in smooth four manifolds. The expression of the representation of a number as a sum of nonvanishing squares is given in terms of the representations of a number as a sum of squares. Using the number theo…

We study surfaces in Euclidean space constructed by the sum of two curves or that are graphs of the product of two functions. We consider the problem to determine all these surfaces with constant Gauss curvature. We extend the results to non degenerate surfaces in Lorentz-Minkowski space.

Study of loops in sums of Laplace eigenfunctions on surfaces.

Study open 3-manifolds as sums of closed ones, finding a classification.

We solve a conjecture of Morgan and Szabo (Embedded genus 2 surfaces in four-manifolds, Preprint) about the relationship of the basic classes of two four-manifolds $X_i$ of simple type with $b_1=0$, $b^+>1$, such that there are embedded Riemann surfaces of genus $g \geq 2$ and self-intersection zero (and representing o…

We compute the genus zero family Gromov-Witten invariants for K3 surfaces using the topological recursion formula and the symplectic sum formula for a degeneration of elliptic K3 surfaces. In particular we verify the Yau-Zaslow formula for non-primitive classes of index two.

Paper analyzes origami slope gaps and their distribution, finding a unique pattern.