Tao Li
We show that given a trivalent graph in $S^3$, either the graph complement contains an essential almost meridional planar surface or thin position for the graph is also bridge position. This can be viewed as an extension of a theorem of Thompson to graphs. It follows that any graph complement always contains a useful planar surface.
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Abby Thompson proved that if a link $K$ is in thin position but not in bridge position then the knot complement contains an essential meridional planar surface, and she asked whether some thin level surface must be essential. This note is to give a positive answer to this question, showing that the if a link is in thin position but not bridge position then a thinnest level surface is essential. A theorem of Rieck and Sedgwick follows as a consequence, which says that thin position of a connected sum of small knots comes in the obvious way.
The self intersection of an immersion i : S^2 \to R^3 dissects S^2 into pieces which are planar surfaces (unless i is an embedding). In this work we determine what collections of planar surfaces may be obtained in this way. In particular, for every n we construct an immersion i : S^2 \to R^3 with 2n triple points, for which all pieces are discs.
We study trivalent graphs in $S^{3}$ whose closed complement is a genus two handlebody. We show that such a graph, when put in thin position, has a simple (i. e. non-loop) level edge.
Let L be a link in the 3-sphere that is in thin position but not in bridge position and let P be a thin level sphere. We generalize a result of Wu by giving a bound on the number of disjoint irreducible compressing disks that P can have, including identifying thin spheres with unique compressing disks. We also give conditions under which P must be incompressible on a particular side or be weakly incompressible. If P is strongly compressible we describe how a pair of compressing disks must lie relative to the link.
A planar portrait of a manifold is the pair of the image and the critical values of the manifold through a stable map into the plane. It can be considerd a geometric representation of the manifold drawn in the plane. The cusped fan is its basic local configuration. In this article, we focus on the fibreing structure over the cusped fan, and give its characterisation. As application, planar portraits of the real, complex, and quaternion projective plane, regular toric surfaces, and some sphere bundles over spheres etc. are constructed. Conversely, the source manifolds of certain planar portrait…
We characterize those planar Peano continua that are homotopy equivalent to 1-dimensional sets. While many planar Peano continua are not homotopically 1-dimensional, we prove that each has fundamental group that embeds in the fundamental group of a 1-dimensional planar Peano continuum. We leave open the following question: Is a planar Peano continuum homotopically 1- dimensional if its fundamental group is isomorphic with the fundamental group of a 1-dimensional planar Peano continuum?
In this note we introduce the (homologically essential) arc complex of a surface as a tool for studying properties of open book decompositions and contact structures. After characterizing destabilizability in terms of the essential translation distance of the monodromy of an open book we given an application of this result to show that there are planer open books of the standard contact structure on the 3-sphere with 5 (or any number larger than 5) boundary components that do not destabilize. We also show that any planar open book with 4 or fewer boundary components does destabilize.
Wu has shown that if a link or a knot $L$ in $S^3$ in thin position has thin spheres, then the thin sphere of lowest width is an essential surface in the link complement. In this paper we show that if we further assume that $L \subset S^3$ is prime, then the thin sphere of lowest width also does not have any vertical cut-disks. We also prove the result for a specific kind of tangles in $S^2 \times [-1,1]$.
Planar Equivalence of Knotoids and Quandle Invariants
针对平面纽结类分类难的问题，本文用受限广群着色与上循环不变量研究其等价性，还提出更强的三角广群上循环不变量，可用于检测平面性及分析对称性。
A Seifert algorithm for integral homology spheres
本文将经典纽结理论中的 Seifert 算法推广到任意闭积分同调 3-球面，利用 Heegaard 分裂的平面图表示和有向缠结的正则投影，构造出 Seifert 圆及半扭转带，从而证明每个有向链环都是有向嵌入曲面的边界。
We construct a Seifert surface for a given null-homologous transverse link in a contact manifold that is compatible with a planar open book decomposition, then obtain a formula of the self-linking number. It extends Bennequin's self-linking number formula for braids in the standard contact 3-sphere.
We consider compact 3-manifolds M having a submersion h to R in which each generic point inverse is a planar surface. The standard height function on a submanifold of the 3-sphere is a motivating example. To (M, h) we associate a connectivity graph G. For M in the 3-sphere, G is a tree if and only if there is a Fox reimbedding of M which carries horizontal circles to a complete collection of complementary meridian circles. On the other hand, if the connectivity graph of the complement of M is a tree, then there is a level-preserving reimbedding of M so that its complement is a connected sum of…
In this paper we generate and systematically classify all prime planar knotoids with up to 5 crossings. We also extend the existing list of knotoids in $S^2$ and add all knotoids with 6 crossings.
A regular circle-valued Morse function on the knot complement C(K) = S^3\K is a function f from C(K) to S^1 which separates critical points and which behaves nicely in a neighborhood of the knot. Such a function induces a handle decomposition on the knot exterior E(K) = S^3\N (K), with the property that every regular level surface contains a Seifert surface for the knot. We rearrange the handles in such a way that the regular surfaces are as simple as possible. To make this precise the concept of circular width for E(K) is introduced. When E(K) is endowed with a handle decomposition which real…
Incompressible planar surfaces in hyperbolic link exteriors in the 3-sphere
对于每个整数N≥3，构造N分支双曲链的例子，其外部包含一个不可压缩的跨越平面曲面，该曲面在每个边界环面上具有非经向且非整数斜率，从而反驳了Eudave-Muñoz和Ozawa的猜想。
Let $f:S^2\to S^2$ be an orientation-preserving branched covering map of degree $d\geq 2$, and let $Σ$ be an oriented Jordan curve passing through the critical values of $f$. Then $Γ:=f^{-1}(Σ)$ is an oriented graph on the sphere. In a group email discussion in Fall 2010, W. Thurston introduced balanced planar graphs and showed that they combinatorially characterize all such $Γ$, where $f$ has $2d-2$ distinct critical values. We give a detailed account of this discussion, along with some examples and an appendix about Hurwitz numbers.
We construct a family of Stein fillable contact homology 3-spheres such that each contact structure of the family is supported by an open book with planar page, and a Stein filling of the contact manifold is of Mazur type.
Brunnian planar braids and simplicial groups
本文研究Brunnian孪生群及其推广，证明多于两链的Brunnian孪生群是自由群，给出2-球面Brunnian doodle成为Brunnian孪生闭包的充要条件，并探讨纯孪生群的单纯结构以组合描述2-球面同伦群。
For a geometrically finite group Gamma of G=SO(n,1), we survey recent developments on counting and equidistribution problems for orbits of Gamma in a homogeneous space H\G where H is trivial, symmetric or horospherical. Main applications are found in an affine sieve on orbits of thin groups as well as in sphere counting problems for sphere packings invariant under a geometrically finite group. In our sphere counting problems, spheres can be ordered with respect to a general conformal metric.
Thin links and Conway spheres
本文提出缠结细性的相对版本，并借助 HFT 与 \widetilde{Kh} 浸入曲线不变量，通过沿 Conway 球面的缠结分解刻画细链环。
We find bounds for the length of the systole -- the shortest essential, non-peripheral closed curve -- for arithmetic punctured spheres with $n$ cusps, for $n=4$ through $n=12$, some of which were previously known due to Schmutz. This is shown using a correspondence between such surfaces and planar triangulations. We show that for $n=7,10,11$, arithmetic surfaces do not achieve the maximal systole length.
Random punctured hyperbolic surfaces & the Brownian sphere
研究按Weil-Petersson测度采样的n+1个穿孔0类随机双曲面Sₙ，证明度量经n⁻¹/⁴缩放后收敛于布朗球面，未缩放时局部收敛于特定无限体积双曲面，通过平面树编码等技术实现。
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