Thin position in the theory of classical knots

Constructing thin subgroups commensurable with the figure-eight knot group

In this paper we find infinitely many lattices in $SL(4,\mathbb{R})$ each of which contains thin subgroups commensurable with the figure-eight knot group.

Non-minimal bridge positions of torus knots are stabilized

We show that any non-minimal bridge decomposition of a torus knot is stabilized and that $n$-bridge decompositions of a torus knot are unique for any integer $n$. This implies that a knot in a bridge position is a torus knot if and only if there exists a torus containing the knot such that it intersects the bridge sphere in two essential loops.

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 for knots, links, and graphs in 3-manifolds

We define a new notion of thin position for a graph in a 3-manifold which combines the ideas of thin position for manifolds first originated by Scharlemann and Thompson with the idea of thin position for knots first originated by Gabai. This thin position has the property that connect summing annuli and pairs-of-pants show up as thin levels. In a forthcoming paper, this new thin position allows us to define two new families of invariants of knots, links, and graphs in 3-manifolds. The invariants in one family are similar to bridge number and the invariants in the other family are similar to Ga…

Cables of thin knots and bordered Heegaard Floer homology

We use bordered Floer homology to give a formula for the knot Floer homology of any (p, pn+1)-cable of a thin knot K in terms of Delta_K(t), tau(K), p, and n. We also give a formula for the Ozsvath-Szabo concordance invariant tau(K_{p, pn+1}) in terms of tau(K), p, and n, and a formula for tau(K_{p,q}) for almost all relatively prime p and q.

Seifert fibered surgeries on strongly invertible knots without primitive/Seifert positions

We find an infinite family of Seifert fibered surgeries on strongly invertible knots which do not have primitive/Seifert positions. Each member of the family is obtained from a trefoil knot after alternate twists along a pair of seiferters for a Seifert fibered surgery on a trefoil knot.

薄厚克莱因瓶之间的最小cobordism

Minimal cobordisms between thin and thick torus knots

研究确定了薄厚克莱因瓶结之间的最小cobordism距离，并揭示了克莱因瓶结作为近最小cobordism横截面的特性。

A locally minimal, but not globally minimal bridge position of a knot

We give a locally minimal, but not globally minimal bridge position of a knot, that is, an unstabilized, nonminimal bridge position of a knot. It implies that a bridge position cannot always be simplified so that the bridge number monotonically decreases to the minimal.

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…

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 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.

Non-parallel essential surfaces in knot complements

We show that if a knot or link has n thin levels when put in thin position then its exterior contains a collection of n disjoint, non-parallel, planar, meridional, essential surfaces. A corollary is that there are at least n/3 tetrahedra in any triangulation of the complement of such a knot.

Morse position of knots and closed incompressible surfaces

In this paper, we study on knots and closed incompressible surfaces in the 3-sphere via Morse functions. We show that both of knots and closed incompressible surfaces can be isotoped into a "related Morse position" simultaneously. As an application, we have following results. *Smallness of Montesinos tangles with length two and Kinoshita's theta curve *Classification of closed incompressible and meridionally incompressible surfaces in 2-bridge theta-curve and handcuff graph complements and the complements of links which admit Hopf tangle decompositions.

Tables of quasi-alternating knots with at most 12 crossings

We are giving tables of quasi-alternating knots with $8\le n \le 12$ crossings. As the obstructions for a knot to be quasialternating we used homology thickness with regards to Khovanov homology, odd homology, and Heegaard-Floer homology $\widehat{HFK}$. Except knots which are homology thick, so cannot be quasialternating, by using the results of our computations [JaSa1], for one of knots which are homology thin, knot $11n_{50}$, J. Greene proved that it is not quasi-alternating, so this is the first example of homologically thin knot which is not quasi-alternating [Gr]. In this paper we provi…

The Khovanov homology of (p,-p,q) pretzel knots

In this paper, we compute the Khovanov homology over \Q for (p,-p,q) pretzel knots for odd values of p from 3 to 15 and arbitrarily large q. We provide a conjecture for the general form of the Khovanov homology of (p,-p,q) pretzel knots. These computations reveal that these knots have thin Khovanov homology (over \Q and \Z). Because Greene has shown that these knots are not quasi-alternating, this provides an infinite class of non-quasi-alternating knots with thin Khovanov homology.

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.

Distance and bridge position

J. Hempel's definition of the distance of a Heegaard surface generalizes to a complexity for a knot which is in bridge position with respect to a Heegaard surface. Our main result is that the distance of a knot in bridge position is bounded above by twice the genus, plus the number of boundary components, of an essential surface in the knot complement. As a consequence knots constructed via sufficiently high powers of pseudo-Anosov maps have minimal bridge presentations which are thin.

Thin presentation of knots and lens spaces

This paper concerns thin presentations of knots K in closed 3-manifolds M^3 which produce S^3 by Dehn surgery, for some slope gamma. If M does not have a lens space as a connected summand, we first prove that all such thin presentations, with respect to any spine of M have only local maxima. If M is a lens space and K has an essential thin presentation with respect to a given standard spine (of lens space M) with only local maxima, then we show that K is a 0-bridge or 1-bridge braid in M; furthermore, we prove the minimal intersection between K and such spines to be at least three, and finally…

Braid group and leveling of a knot

Any knot $K$ in genus-$1$ $1$-bridge position can be moved by isotopy to lie in a union of $n$ parallel tori tubed by $n-1$ tubes so that $K$ intersects each tube in two spanning arcs, which we call a leveling of the position. The minimal $n$ for which this is possible is an invariant of the position, called the level number. In this work, we describe the leveling by the braid group on two points in the torus, which yields a numerical invariant of the position, called the $(1, 1)$-length. We show that the $(1, 1)$-length equals the level number. We then find braid descriptions for $(1,1)$-posi…

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。

Invisible knots and rainbow rings: knots not determined by their determinants

We determine p-colorability of the paradromic rings. These rings arise by generalizing the well-known experiment of bisecting a Mobius strip. Instead of joining the ends with a single half twist, use $m$ twists, and, rather than bisecting ($n = 2$), cut the strip into $n$ sections. We call the resulting collection of thin strips $P(m,n)$. By replacing each thin strip with its midline, we think of $P(m,n)$ as a link, that is, a collection of circles in space. Using the notion of $p$-colorability from knot theory, we determine, for each $m$ and $n$, which primes $p$ can be used to color $P(m,n)$…

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…

Counting Generic Quadrisecants of Polygonal Knots

Let $K$ be a polygonal knot in general position with vertex set $V$. A \emph{generic quadrisecant} of $K$ is a line that is disjoint from the set $V$ and intersects $K$ in exactly four distinct points. We give an upper bound for the number of generic quadrisecants of a polygonal knot $K$ in general position. This upper bound is in terms of the number of edges of $K$.

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…

具有(1,1)-模式的卫星纽结的一些三维与四维不变量

Some Three and Four-Dimensional Invariants of Satellite Knots with (1,1)-Patterns

本文利用带边Floer同调的浸入曲线解释，研究具有(1,1)-三叶结模式的卫星纽结，计算其三维与四维不变量，并证明相关纤维化、转向单值及非薄Floer同调性质。

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…

Thin position and essential planar surfaces

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.

Unperturbed weakly reducible non-minimal bridge positions

A bridge position of a knot is said to be perturbed if there exists a cancelling pair of bridge disks. Motivated by the examples of knots admitting unperturbed strongly irreducible non-minimal bridge positions due to Jang-Kobayashi-Ozawa-Takao, we derive examples of unperturbed weakly reducible non-minimal bridge positions. Also, a bridge version of Gordon's Conjecture is proposed: the connected sum of unperturbed bridge positions is unperturbed.

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.

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.

3-流形嵌入5球面中的桥位置

Bridge position of 3-manifolds embedded in the 5-sphere

研究3-流形在5球面中的桥分解，推广经典结在3球面中的桥位置及Meier和Zupan在4球面中的桥三分割方法，证明任意嵌入3-流形均可分解，且以四个平凡打结图表示，举例包括S²打结和ribbon 3-结。

利用 Plat 位置生成无穷多个双曲纽结

Generating Infinitely Many Hyperbolic Knots with Plats

本文借助 plat 位置、辫群动力学与二重分支覆盖的 Heegaard 距离，利用 pseudo-Anosov 辫构造无穷多互不相同的素双曲 n-桥纽结，并给出亏格与体积的线性下界。

The Knotting-Unknotting Game played on Sums of Rational Shadows

We consider the recently introduced knotting-unknotting game, in which two players take turns resolving crossings in a knot diagram which initially is missing all its crossing information. Once the knot is fully resolved, the winner is decided by whether the knot is equivalent to the unknot. In this paper we determine which player wins under perfect play in the initial positions (knot shadows) that are guaranteed to produce rational knots or sums of rational knots. This is accomplished in part by an analysis of the effects of pseudo Reidemeister moves on positions, and a computer verification …

Reduction of bridge positions along a bridge disk

Suppose a knot in a $3$-manifold is in $n$-bridge position. We consider a reduction of the knot along a bridge disk $D$ and show that the result is an $(n-1)$-bridge position if and only if there is a bridge disk $E$ such that $(D, E)$ is a cancelling pair. We apply this to an unknot $K$, in $n$-bridge position with respect to a bridge sphere $S$ in the $3$-sphere, to consider the relationship between a bridge disk $D$ and a disk in the $3$-sphere that $K$ bounds. We show that if a reduction of $K$ along $D$ yields an $(n-1)$-bridge position, then $K$ bounds a disk that contains $D$ as a subdi…

Semisimple tunnels

A knot in the 3-sphere in genus-1 1-bridge position (called a (1,1)-position) can be described by an element of the braid group of two points in the torus. Our main results tell how to translate between a braid group element and the sequence of slope invariants of the upper and lower tunnels of the (1,1)-position. After using them to verify previous calculations of the slope invariants for all tunnels of 2-bridge knots and (1,1)-tunnels of torus knots, we obtain characterizations of the slope sequences of tunnels of 2-bridge knots, and of a class of tunnels we call toroidal. The main results l…

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.

Bridge numbers for virtual and welded knots

Using Gauss diagrams, one can define the virtual bridge number ${\rm vb}(K)$ and the welded bridge number ${\rm wb}(K),$ invariants of virtual and welded knots with ${\rm wb}(K) \leq {\rm vb}(K).$ If $K$ is a classical knot, Chernov and Manturov showed that ${\rm vb}(K) = {\rm br}(K),$ the bridge number as a classical knot, and we ask whether the same thing is true for welded knots. The welded bridge number is bounded below by the meridional rank of the knot group $G_K$, and we use this to relate this question to a conjecture of Cappell and Shaneson. We show how to use other virtual and welded…

Seifert fibered surgeries with distinct primitive/Seifert positions

We call a pair (K, m) of a knot K in the 3-sphere S^3 and an integer m a Seifert fibered surgery if m-surgery on K yields a Seifert fiber space. For most known Seifert fibered surgeries (K, m), K can be embedded in a genus 2 Heegaard surface of S^3 in a primitive/Seifert position, the concept introduced by Dean as a natural extension of primitive/primitive position defined by Berge. Recently Guntel has given an infinite family of Seifert fibered surgeries each of which has distinct primitive/Seifert positions. In this paper we give yet other infinite families of Seifert fibered surgeries with …

Bridge position and the representativity of spatial graphs

First, we extend Otal's result for the trivial knot to trivial spatial graphs, namely, we show that for any bridge tangle decomposing sphere $S^2$ for a trivial spatial graph $Γ$, there exists a 2-sphere $F$ such that $F$ contains $Γ$ and $F$ intersects $S^2$ in a single loop. Next, we introduce two invariants for spatial graphs. As a generalization of the bridge number for knots, we define the {\em bridge string number} $bs(Γ)$ of a spatial graph $Γ$ as the minimal number of $|Γ\cap S^2|$ for all bridge tangle decomposing sphere $S^2$. As a spatial version of the representativity for a graph …

Nimble evolution for pretzel Khovanov polynomials

We conjecture explicit evolution formulas for Khovanov polynomials for pretzel knots in some regions in the windings space. Our description is exhaustive for genera 1 and 2. As previously observed, evolution at T != -1 is not fully smooth: it switches abruptly at the boundaries between different regions. We reveal that this happens also at the boundary between thin and thick knots, moreover, the thick-knot domain is further stratified. For thin knots evolution is governed by the standard T-deformation lambda of the eigenvalues of the R-matrix. Emerging in the thick knots regions are additional…

Thin Position for Tangles

If a tangle, K, in the 3-ball has no planar, meridional, essential surfaces in its exterior then thin position for K has no thin levels.

A search method for thin positions of links

We give a method for searching for thin positions of a given link.

Thin position and planar surfaces for graphs in the 3-sphere

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.

Arc distance equals level number

A knot K in 1-bridge position with respect to a genus-g Heegaard surface in a 3-manifold can be moved by isotopy through knots in 1-bridge position until it lies in a union of n parallel genus-g surfaces tubed together by n-1 straight tubes, with K intersecting each tube in two arcs connecting the ends. We prove that the minimum n for which this is possible is equal to a Hempel-type distance invariant defined using an arc complex of the two holed genus-g surface

一个能解开结的机器人

A robot that unknots knots

该研究提出一个能解开结的机器人模型，通过组合方法证明了升序或降序结图可转换为零交叉无结图，并给出了 Reidemeister 移动次数的上界及新的移动序列证明。

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…

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.

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

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…

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…

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。

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 …

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.