Related papers: Toroidal and Klein bottle boundary slopes

We prove that 1) There exist infinitely many non-trivial codimension one "thick" knots in $\mathbb{R}^5$; 2) For each closed four-dimensional smooth manifold $M$ and for each sufficiently small positive $\epsilon$ the set of isometry…

Recently Penskoi [J. Geom. Anal. 25 (2015), 2645-2666, arXiv:1308.1628] generalized the well known two-parametric family of Lawson tau-surfaces $\tau_{r,m}$ minimally immersed in spheres to a three-parametric family $T_{a,b,c}$ of tori and…

Let $T$ be a graph in a compact, orientable 3--manifold $M$ and let $\Gamma$ be a subgraph. $T$ can be placed in bridge position with respect to a Heegaard surface $H$. We show that if $H$ is what we call $(T,\Gamma)$-c-weakly reducible in…

Let (V,W;F) be a weakly reducible, unstabilized, genus three Heegaard splitting in an orientable, irreducible 3-manifold M. In this article, we prove that either the disk complex D(F) is contractible or F is critical. Hence, the topological…

Main Result: Let $(M,L)$ be a smooth complex polarized threefold. Then the linear system $| K+tL|$ separates any two different points on $M$ for any $t\ge 6$, where $K$ is the canonical bundle of $M$. The argument in the proof is a variant…

A knot type is exchange reducible if an arbitrary closed n-braid representative can be changed to a closed braid of minimum braid index by a finite sequence of braid isotopies, exchange moves and +/- destabilizations. In the manuscript [J…

Suppose $M$ is a hyperbolic 3-manifold which admits two Dehn fillings $M(r_1)$ and $M(r_2)$ such that $M(r_1)$ contains an essential torus and $M(r_2)$ contains an essential annulus. It is known that $\Delta = \Delta(r_1, r_2) \leq 5$. We…

Extending work of Kapouleas and Yang, for any integers $N \geq 2$, $k, \ell \geq 1$, and $m$ sufficiently large, we apply gluing methods to construct in the round $3$-sphere a closed embedded minimal surface that has genus $k\ell…

Suppose that $M$ is a compact, connected three-manifold with boundary. We show that if the universal cover has infinitely many boundary components then $M$ has an ideal triangulation which is essential: no edge can be homotoped into the…

The non-orientable 4-genus of a knot $K$ in $S^{3}$, denoted $\gamma_4(K)$, measures the minimum genus of a non-orientable surface in $B^{4}$ bounded by $K$. We compute bounds for the non-orientable 4-genus of knots $T_{5, q}$ and $T_{6,…

Thurston's ending lamination conjecture proposes that a finitely generated Kleinian group is uniquely determined (up to isometry) by the topology of its quotient and a list of invariants that describe the asymptotic geometry of its ends. We…

For each integer m>1 and l>0 we construct a pair of compact embedded minimal surfaces of genus 1+4m(m-1)l. These surfaces desingularize the m Clifford tori meeting each other along a great circle at the angle of \pi/m. They are invariant…

We show that there exist infinitely many closed contact 3-manifolds containing half Giroux torsion along a separating torus whose contact invariants do not vanish. This provides counterexamples to Ghiggini's conjecture and suggests that…

It is a basic question in contact geometry to classify all non-isotopic tight contact structures on a given 3-manifold. If the manifold has a boundary, we need also specify the dividing set on the boundary. In this paper, we answer the…

A classic result of Shi and Tam states that a 2-sphere of positive Gauss and mean curvature bounding a compact 3-manifold with nonnegative scalar curvature, must have total mean curvature not greater than that of the isometric embedding…

Graphs that are critical (minimal excluded minors) for embeddability in surfaces are studied. In Part I, it was shown that graphs that are critical for embeddings into surfaces of Euler genus $k$ or for embeddings into nonorientable surface…

The partition number $\pi(K)$ of a simplicial complex $K\subset 2^{[m]}$ is the minimum integer $\nu$ such that for each partition $A_1\uplus\ldots\uplus A_\nu = [m]$ of $[m]$ at least one of the sets $A_i$ is in $K$. A complex $K$ is…

In this paper, we prove that the trisection genus of the Akbulut cork is $3$ and construct infinitely many corks with trisection genus $3$. These results give the first examples of contractible $4$-manifolds whose trisection genera are…

Suppose $M$ is a manifold with boundary. Choose a point $o\in\partial M$. We investigate the prescribed Ricci curvature equation $\Ric(G)=T$ in a neighborhood of $o$ under natural boundary conditions. The unknown $G$ here is a Riemannian…

We develop obstructions to a knot K in the 3-sphere bounding a smooth punctured Klein bottle in the 4-ball. The simplest of these is based on the linking form of the 2-fold branched cover of the 3-sphere branched over K. Stronger…