A search method for thin positions of links

Compressing thin spheres in the complement of a link

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.

Thin position for a connected sum of small knots

We show that every thin position for a connected sum of small knots is obtained in an obvious way: place each summand in thin position so that no two summands intersect the same level surface, then connect the lowest minimum of each summand to the highest maximum of the adjacent summand below.

Thin position in the theory of classical knots

This is a chapter that is to appear in the "Handbook of Knot Theory", edited by William W. Menasco and Morwen B. Thistlethwaite.

Non-minimal bridge position of $2$-cable links

Suppose that every non-minimal bridge position of a knot $K$ is perturbed. We show that if $L$ is a $(2, 2q)$-cable link of $K$, then every non-minimal bridge position of $L$ is also perturbed.

A random link via bridge position is hyperbolic

We show that a random link defined by random bridge splitting is hyperbolic with asymptotic probability 1.

Thin position for incompressible surfaces in 3-manifolds

In this paper, we give an algorithm to build all compact orientable atoroidal Haken 3-manifolds with tori boundary or closed orientable Haken 3-manifolds, so that in both cases, there are embedded closed orientable separating incompressible surfaces which are not tori. Next, such incompressible surfaces are related to Heegaard splittings. For simplicity, we focus on the case of separating incompressible surfaces, since non-separating ones have been extensively studied. After putting the surfaces into Morse position relative to the height function associated to the Heegaard splittings, a thin p…

Cut-disks for level spheres in link and tangle complements

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 $S^2 \times [-1,1]$.

A note on the nearly additivity of knot width

Let k be a knot in S3. In [8], H.N. Howards and J. Schultens introduced a method to construct a manifold decomposition of double branched cover of (S3, k) from a thin position of k. In this article, we will prove that if a thin position of k induces a thin decomposition of double branched cover of (S3,k) by Howards and Schultens' method, then the thin position is the sum of prime summands by stacking a thin position of one of prime summands of k on top of a thin position of another prime summand, and so on. Therefore, k holds the nearly additivity of knot width (i.e. for k = k1#k2, w(k) = w(k1…

Thin Position through the lens of trisections of 4-manifolds

Motivated by M. Scharlemann and A. Thompson's definition of thin position of 3-manifolds, we define the width of a handle decomposition a 4-manifold and introduce the notion of thin position of a compact smooth 4-manifold. We determine all manifolds having width equal to $\{1,\dots, 1\}$, and give a relation between the width of $M$ and its double $M\cup_{id_\partial} \overline M$. In particular, we describe how to obtain genus $2g+2$ and $g+2$ trisection diagrams for sphere bundles over orientable and non-orientable surfaces of genus $g$, respectively. By last, we study the problem of describ…

Thin position for knots in a 3-manifold

We extend the notion of thin multiple Heegaard splittings of a link in a 3-manifold to take into consideration not only compressing disks but also cut-disks for the Heegaard surfaces. We prove that if H is a c-strongly compressible bridge surface for a link K contained in a closed orientable irreducible 3-manifold M then one of the following is satisfied: H is stabilized, H is meridionally stabilized, H is perturbed, a component of K is removable, M contains an essential meridional surface.

无法被单调简化的链环桥位

Bridge Positions of Links That Cannot Be Monotonically Simplified

针对任意 3<m<n，本文利用 Takao 等人的判据构造无限链环族，证明存在局部极小但非全局极小的未扰动 n-桥位，且简化到 m-桥位前必须先经历扰动。

纽结的桥位置与平台表示

Bridge Positions and Plat Presentations of Links

本文建立纽结的桥位置与平台表示的Hilden双陪集类的对应关系，将平台表示的代数问题转化为桥位置的几何问题，重新确立相关结果并探讨其在辫子平台闭包研究中的应用。

表面在任意3流形中链补集的标准位置

Standard position for surfaces in link complements in arbitrary 3-manifolds

本文证明了表面的标准位置可扩展至更广泛的链接类，包括弱通用交替链接，并展示了此类链接的素性及Conway球体与图样的交互特性。

Quasi-alternating links and odd homology: computations and conjectures

We present computational results about quasi-alternating knots and links and odd homology obtained by looking at link families in the Conway notation. More precisely, we list quasi-alternating links up to 12 crossings and the first examples of quasi-alternating knots and links with at least two different minimal diagrams, where one is quasi-alternating and the other is not. We provide examples of knots and links with $n\le 12$ crossings which are homologically thin and have no minimal quasi-alternating diagrams. These links are candidates for homologically thin links that are not quasi-alterna…

Thin Tree Position

We introduce a method for creating a special type of tree, called a tree position, from a weighted graph. Leaves of the tree correspond to vertices of the original graph, and the tree edges contain information which can be used to partition these vertices. By repeatedly applying reducing operations to the tree position we arrive at a special type of tree position we call thin, and we show that partitions arising from thin tree positions have especially nice properties. The algorithm is based the topological notion of thin position for knots and 3-manifolds and builds on the previously defined …

细戈尔迪安解链

Thin Gordian Unlinks

本文首次构造出戈尔迪安解链的例子，并引入厚度在[1,2)的单参数细纽结模型，证明更细的管状邻域会产生不同于经典粗纽结的绳结几何。

Circular thin position for knots in the 3-sphere

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…

通过薄位置证明强哈肯定理

Strong Haken via Thin Position

本文利用 Heegaard 分裂的薄位置给出哈肯引理和强哈肯定理的新证明，并重新表述 Casson–Gordon 定理，揭示弱可约 Heegaard 分裂关于不可压缩曲面的额外结构。

alpha-sloped Generalized Heegaard splittings of 3-manifolds

We generalize the definition of thin position of Scharlemann and Thompson for compact orientable 3-manifolds with torus boundary components and introduce $α$-sloped generalized Heegaard splittings. We examine its relationship to generalized Heegaard splittings of manifolds resulting from Dehn filling. We compare alpha-sloped thin position of 3-manifolds to other types of thin position for knots and 3-manifolds and discuss how this kind of decomposition gives an organic picture of M and allows the structure of the manifold to dictate the most natural slope(s) on the boundary. Additionally, we p…

Khovanov 同调中细区域上的挠部分

Torsion in thin regions of Khovanov homology

本文证明 Khovanov 同调在局部同调分次范围内满足细性条件时只含 Z₂ 挠，并构造满足条件的无穷族 3-辫，部分证实 3-辫仅有 Z₂ 挠的猜想。

Properties of knots preserved by cabling

We examine geometric properties of a knot J that are unchanged by taking a (p,q)-cable K of J. Specifically, we relate w(K) to w(J), where w(K) is the width of K in the sense of Gabai. We use this information to demonstrate that thin position is a minimal bridge position of J if and only if the same is true for K, and more generally we show that any thin position of K is an "obvious" cabling of a thin position of J. We conclude by proving that J is meridionally small (mp-small) if and only if K is meridionally small (mp-small), and if J is mp-small and every non-minimal bridge position of J is…

Heegaard Splittings with Boundary and Almost Normal Surfaces

This paper generalizes the definition of a Heegaard splitting to unify Scharlemann and Thomspon's concept of thin position for 3-manifolds, Gabai's thin position for knots, and Rubinstein's almost normal surface theory. This gives generalizations of theorems of Scharlemann, Thompson, Rubinstein, and Stocking. In the final section, we use this machinery to produce an algorithm to determine the bridge number of a knot, provided thin position for the knot coincides with bridge position. We also present several finiteness and algorithmic results about Dehn fillings with "small" Heegaard genus.

Knots with compressible thin levels

We produce embeddings of knots in thin position that admit compressible thin levels. We also find the bridge number of tangle sums where each tangle is high distance.

Topological graph clustering with thin position

A clustering algorithm partitions a set of data points into smaller sets (clusters) such that each subset is more tightly packed than the whole. Many approaches to clustering translate the vector data into a graph with edges reflecting a distance or similarity metric on the points, then look for highly connected subgraphs. We introduce such an algorithm based on ideas borrowed from the topological notion of thin position for knots and 3-dimensional manifolds.

细链环与 Conway 球面

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。

Twisted Alexander polynomials of torus links

In this paper we give an explicit formula for the twisted Alexander polynomial of any torus link and show that it is a locally constant function on the $SL(2, \mathbb C)$-character variety. We also discuss similar things for the higher dimensional twisted Alexander polynomial and the Reidemeister torsion.

Torsion in Khovanov homology of homologically thin knots

We prove that every $\mathbb{Z}_2$H-thin link has no $2^k$-torsion for $k>1$ in its Khovanov homology. Together with previous results by Eun Soo Lee and the author, this implies that integer Khovanov homology of non-split alternating links is completely determined by the Jones polynomial and signature. Our proof is based on establishing an algebraic relation between Bockstein and Turner differentials on Khovanov homology over $\mathbb{Z}_2$. We conjecture that a similar relation exists between the corresponding spectral sequences.

Levelling an unknotting tunnel

It is a consequence of theorems of Gordon-Reid [Tangle decompositions of tunnel number one knots and links, J. Knot Theory and its Ramifications, 4 (1995) 389-409] and Thompson [Thin position and bridge number for knots in the 3-sphere, Topology, 36 (1997) 505-507] that a tunnel number one knot, if put in thin position, will also be in bridge position. We show that in such a thin presentation, the tunnel can be made level so that it lies in a level sphere. This settles a question raised by Morimoto [A note on unknotting tunnels for 2-bridge knots, Bulletin of Faculty of Engineering Takushoku U…

A lower bound on the width of satellite knots

Thin position for knots in the 3-sphere was introduced by Gabai and has been used in a variety of contexts. We conjecture an analogue to a theorem of Schubert and Schultens concerning the bridge number of satellite knots. For a satellite knot K, we use the companion torus T to provide a lower bound for w(K), proving the conjecture for K with a 2-bridge companion. As a corollary, we find thin position for any satellite knot with a braid pattern and 2-bridge companion.

On the Khovanov and knot Floer homologies of quasi-alternating links

Quasi-alternating links are a natural generalization of alternating links. In this paper, we show that quasi-alternating links are "homologically thin" for both Khovanov homology and knot Floer homology. In particular, their bigraded homology groups are determined by the signature of the link, together with the Euler characteristic of the respective homology (i.e. the Jones or the Alexander polynomial). The proofs use the exact triangles relating the homology of a link with the homologies of its two resolutions at a crossing.