横突结节

结和接触几何形状

Knots and Contact Geometry

我们对勒让德环面纽结进行了分类，并在3-球面上的紧切触结构中画出了8个纽结，直到勒让德合痕。作为一个推论，我们也得到了横向环面结和数字8结的横向合痕的分类。

Torus knots obtained by negatively twisting torus knots

Twisted torus knots are torus knots with some full twists added along some number of adjacent strands. There are infinitely many known examples of twisted torus knots which are actually torus knots. We give eight more infinite families of such twisted torus knots with a single negative twist.

Concordances from differences of torus knots to $L$-space knots

It is known that connected sums of positive torus knots are not concordant to $L$-space knots. Here we consider differences of torus knots. The main result states that the subgroup of the concordance group generated by two positive torus knots contains no nontrivial $L$-space knots other than the torus knots themselves. Generalizations to subgroups generated by more than two torus knots are also considered.

The AJ-conjecture and cabled knots over torus knots

We show that most cabled knots over torus knots in $S^3$ satisfy the AJ-conjecture, namely each $(r,s)$-cabled knot over each $(p,q)$-torus knot satisfies the $AJ$-conjecture if $r$ is not a number between $0$ and $pqs$.

Concordances from connected sums of torus knots to L-space knots

If a knot is a nontrivial connected sum of positive torus knots, then it is not concordant to an L-space knot.

扭转环面纽结的Alexander多项式

The Alexander polynomial of twisted torus knots

本文提出扭转环面纽结的Alexander多项式的显式公式，利用纽结群表示与Fox微分运算，并应用于亏格下界及识别非L-空间纽结。

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.

Additional cases of positive twisted torus knots

A twisted torus knot is a knot obtained from a torus knot by twisting adjacent strands by full twists. The twisted torus knots lie in $F$, the genus 2 Heegaard surface for $S^3$. Primitive/primitive and primitive/Seifert knots lie in $F$ in a particular way. Dean gives sufficient conditions for the parameters of the twisted torus knots to ensure they are primitive/primitive or primitive/Seifert. Using Dean's conditions, Doleshal shows that there are infinitely many twisted torus knots that are fibered and that there are twisted torus knots with distinct primitive/Seifert representatives with t…

Numerical semigroups, Dedekind sums and Alexander polynomials of torus knots

We connect Dedekind sums and Alexander polynomials of torus knots.

环面结的横截数

Crosscap numbers of torus knots

3-球面中纽结的交叉数是以该纽结为界的非可定向曲面的最小亏格。我们确定环面结的交叉数。

On sums of torus knots concordant to alternating knots

We consider the question, asked by Friedl, Livingston and Zentner, of which sums of torus knots are concordant to alternating knots. After a brief analysis of the problem in its full generality, we focus on sums of two torus knots. We describe some effective obstructions based on Heegaard Floer homology.

Fibered and primitive/Seifert twisted torus knots

The twisted torus knots lie on the standard genus 2 Heegaard surface for $S^3$, as do the primitive/primitive and primitive/Seifert knots. It is known that primitive/primitive knots are fibered, and that not all primitive/Seifert knots are fibered. Since there is a wealth of primitive/Seifert knots that are twisted torus knots, we consider the twisted torus knots to partially answer the question of which primitive/Seifert knots are fibered. A braid computation shows that a particular family of twisted torus knots is fibered, and that computation is then used to generalize the results of a prev…

On composite twisted torus knots

In the present note, we will show that there are infinitely many composite twisted torus knots.

Semigroups of L-space knots and nonalgebraic iterated torus knots

Algebraic knots are known to be iterated torus knots and to admit L-space surgeries. However, Hedden proved that there are iterated torus knots that admit L-space surgeries but are not algebraic. We present an infinite family of such examples, with the additional property that no nontrivial linear combination of knots in this family is concordant to a linear combination of algebraic knots. The proof uses the Ozsvath-Stipsicz-Szabo Upsilon function, and also introduces a new invariant of L-space knots, the formal semigroup.

环面结是Fourier-（1，1，2）结

Torus knots are Fourier-(1,1,2) knots

每个环面结都可以表示为傅立叶-（1，1，2）结，这是这种结的最简单可能的傅立叶表示。这回答了Kauffman的一个问题，证实了Boocher，Daigle，Hoste和Zheng的猜想.特别地，环面结T（p，q）可以被参数化为x（t）=cos（pt），y（t）=cos（qt+pi/（2 p）），以及z（t）=cos（pt+pi/2）\cos（（q-p）t+pi/（2 p）-pi/（4 q））。

环面结的解结序列

Unknotting sequences for torus knots

一个纽结的解结数由它的切片亏格从下而上限定。众所周知，环面纽结的属数和解纽结数是一致的。本文刻画了亏格界为尖的拟正纽结：一个拟正纽结的切片亏格等于它的解纽结数，当且仅当给定的纽结出现在环面纽结的解纽结序列中.

Gordian adjacency for torus knots

A knot K is called Gordian adjacent to a knot L if there exists an unknotting sequence for L containing K. We provide a sufficient condition for Gordian adjacency of torus knots via the study of knots in the thickened torus. We also completely describe Gordian adjacency for torus knots of index 2 and 3 using Levine-Tristram signatures as obstructions to Gordian adjacency. Finally, Gordian adjacency for torus knots is compared to the notion of adjacency for plane curve singularities.

Note on the distortion of (2,q)-torus knots

We show that the distortion of the (2,q)-torus knot is not bounded linearly from below.

Addendum to "Spherical structures on torus knots and links"

The present paper is an addendum to "Spherical structures on torus knots and links", arXiv:1008.0312, and concerns more general case of torus knot and link cone-manifolds.

On the knot Floer homology of twisted torus knots

In this paper we study the knot Floer homology of a subfamily of twisted $(p, q)$ torus knots where $q \equiv\pm1$ (mod $p$). Specifically, we classify the knots in this subfamily that admit L-space surgeries. To do calculations, we use the fact that these knots are $(1, 1)$ knots and, therefore, admit a genus one Heegaard diagram.

无法通过扭转解开的环面结

Torus knots that cannot be untied by twisting

本文给出了一个环面纽结可由一次扭转解开的必要条件。利用这个结果，我们给出了无穷多个不能通过单次扭转解开的环面结。

An Unknotting Sequence for Torus Knots

In this paper, the authors give an unknotting sequence for torus knots and also provide unknotting numbers of $_n14_{17191}, \ _n14_{14274}, \ _n14_{18351}, \ _n14_{24498}$ and some other knots from the knot table of Hoste-Thistlethwite.

On colorability of knots by rotations, Torus knot and PL trochoid

The set consisting of all rotations of the Euclidean plane is equipped with a quandle structure. We show that a knot is colorable by this quandle if and only if its Alexander polynomial has a root on the unit circle in $\mathbb{C}$. Further we enumerate all non-trivial colorings of a torus knot diagram by the quandle using PL trochoids. As an application of these results, we have the complete factorization of the Alexander polynomial of the torus knot.

Non left-orderable surgeries on negatively twisted torus knots

We show that certain negatively twisted torus knots admit Dehn surgeries yielding 3-manifolds with non left-orderable fundamental groups.

Unexpected Essential Surfaces among Exteriors of Twisted Torus Knots

The twisted torus knots K(p, q; r, s) are obtained by performing a sequence of s full twists on r adjacent strands of (p, q)-torus knots. Morimoto asked whether all twisted torus knots with essential tori in the exterior fit into one of two families. We prove that the answer to this question is no, by finding two different new families of toroidal twisted torus knots.

Triple-crossing number, the genus of a knot or link and torus knots

We show that the triple-crossing number of any knot is greater or equal to twice its (canonical) genus and we show an even stronger bound in the case of links. As an application we show that this bound is strong enough to obtain the triple-crossing numbers of all torus knots, and of many more knots and their connected sums.

焊接纽结与带状环面纽结的非平凡性

Non-triviality of welded knots and ribbon torus-knots

本文通过焊接经典纽结图中的交叉点构造焊接纽结，并利用基本群证明其非平凡性，从而给出非平凡带状环面的例子。

On tangle decompositions of twisted torus knots

In the present paper, we will show that for any integer n>0 there are infinitely many twisted torus knots with n-string essential tangle decompositions.

$(n,2)$ -环面纽结和曲线$Y^2=X^3+X^2$的有穷级不变量

Finite order invariants for $(n,2)$-torus knots and the curve $Y^2=X^3+X^2$

我们描述了所有$(n,2)$ -环面纽结集合上的有穷级不变量代数。

Upper bounds for the complexity of torus knot complements

We establish upper bounds for the complexity of Seifert fibered manifolds with nonempty boundary. In particular, we obtain potentially sharp bounds on the complexity of torus knot complements.

无限族非纤维化扭转环形结

Infinite families of non-fibered twisted torus knots

本文通过显式公式构造了无限族非纤维化扭转环形结，证明其亚历山大多项式主系数可任意取整数值。

The minimal sequence of Reidemister moves bringing the diagram of $(n+1,n)$-torus knot to that of $(n,n+1)$-torus knot

Let $D(p,q)$ be the usual knot diagram of the $(p,q)$-torus knot, that is, $D(p,q)$ is the closure of the $p$-braid $(σ_1^{-1} σ_2^{-1}... σ_{p-1}^{-1})^q$. As is well-known, $D(p,q)$ and $D(q,p)$ represent the same knot. It is shown that $D(n+1,n)$ can be deformed to $D(n,n+1)$ by a sequence of $\{(n-1)n(2n-1)/6 \} + 1$ Reidemeister moves, which consists of a single RI move and $(n-1)n(2n-1)/6$ RIII moves. Using cowrithe, we show that this sequence is minimal over all sequences which bring $D(n+1,n)$ to $D(n,n+1)$.

On the signatures of torus knots

We study properties of the signature function of the torus knot $T_{p,q}$. First we provide a very elementary proof of the formula for the integral of the signatures over the circle. We obtain also a closed formula for the Tristram--Levine signature of a torus knot in terms of Dedekind sums.

Uniqueness of higher genus bridge surfaces for torus knots

We show that a torus knot which is not 2-bridge has a unique irreducible bridge splitting of positive genus.

环面结的无向带解结数

The unoriented band unknotting numbers of torus knots

本文研究环面结的无向带解结数，利用无向纽结 Floer 同调的挠阶证明其与可由环面参数计算的 pinch 数相等。

Virtual unknotting numbers of certain virtual torus knots

The virtual unknotting number of a virtual knot is the minimal number of crossing changes that makes the virtual knot to be the unknot, which is defined only for virtual knots virtually homotopic to the unknot. We focus on the virtual knot obtained from the standard (p,q)-torus knot diagram by replacing all crossings on one overstrand into virtual crossings and prove that its virtual unknotting number is equal to the unknotting number of the $(p,q)$-torus knot, i.e. it is (p-1)(q-1)/2.

The maximum Levine-Tristram signature of torus knots

We prove that the maximum of the Levine-Tristram signature function of a torus knot satisfies a reduction formula analogous to a result by Gordon-Litherland-Murasugi for the classical signature.

Petal number of torus knots of type $(r,r+2)$

Let $r$ be an odd integer, $r\ge3$. Then the petal number of the torus knot of type $(r,r+2)$ is equal to $2r+3$.

On the complexity of torus knot recognition

We show that the problem of recognizing that a knot diagram represents a specific torus knot, or any torus knot at all, is in the complexity class ${\sf NP} \cap {\sf co\text{-}NP}$, assuming the generalized Riemann hypothesis. We also show that satellite knot detection is in ${\sf NP}$ under the same assumption, and that cabled knot detection and composite knot detection are unconditionally in ${\sf NP}$. Our algorithms are based on recent work of Kuperberg and of Lackenby on detecting knottedness.

Twisted torus knots T(p,q,3,s) are tunnel number one

We show that twisted torus knots $T(p,q,3,s)$ are tunnel number one. A short spanning arc connecting two adjacent twisted strands is an unknotting tunnel.

The strong AJ conjecture for cables of torus knots

The AJ conjecture, formulated by Garoufalidis, relates the A-polynomial and the colored Jones polynomial of a knot in the 3-sphere. It has been confirmed for all torus knots, some classes of two-bridge knots and pretzel knots, and most cabled knots over torus knots. The strong AJ conjecture, formulated by Sikora, relates the A-ideal and the colored Jones polynomial of a knot. It was confirmed for all torus knots. In this paper we confirm the strong AJ conjecture for most cabled knots over torus knots.

There exist no Minimally Knotted Planar Spatial Graphs on the Torus

We show that all nontrivial embeddings of planar graphs on the torus contain a nontrivial knot or a nonsplit link. This is equivalent to showing that no minimally knotted planar spatial graphs on the torus exist that contain neither a nontrivial knot nor a nonsplit link all of whose components are unknots.

环面纽结族的Whitehead二重体体积猜想的证明

Proof of the volume conjecture for Whitehead doubles of a family of torus knots

提出了一种计算卫星节点的色琼斯多项式的方法，并以节点的怀特黑德对偶为例进行了说明。然后，我们证明了一个家庭的环面结的Whitehead双打的体积猜想，并显示一些有趣的意见。

Kashaev invariants of twice-iterated torus knots

We calculate the asymptotic behavior of the Kashaev invariant of a twice-itarated torus knot and obtain topological interpretation of the formula in terms of the Chern--Simons invariant and the twisted Reidemeister torsion.

环面纽结与Dunwoody流形

Torus knots and Dunwoody manifolds

我们得到一个显式表示，作为邓伍迪流形，所有循环分支覆盖的环面结类型$(p,mp\pm 1)$，与$p>1$和$m>0$。

纺制环面结的分类

The Classification of Spun Torus Knots

S. Satoh已经定义了一种构造，以获得给定焊接结的带状环面结。这种结构被称为满射。我们证明它不是内射的。使用外围结构的不变量，可以提供对注入性的这种失败的限制。特别是，我们还提供了一个代数分类的建设时，仅限于经典的结，它是等价的环面旋转建设。

Genus one cobordisms between torus knots

We determine the pairs of torus knots that have a genus one cobordism between them, with one notable exception. This is done by combining obstructions using $ν^+$ from the Heegaard Floer knot complex and explicit constructions of cobordisms. As an application, we determine the pairs of torus knots related by a single crossing change. Also, we determine the pairs of Thurston-Bennequin number maximizing Legendrian torus knots that have a genus one exact Lagrangian cobordism, with one exception.

An infinite family of Legendrian torus knots distinguished by cube number

For a knot $K$ the cube number is a knot invariant defined to be the smallest $n$ for which there is a cube diagram of size $n$ for $K$. There is also a Legendrian version of this invariant called the \emph{Legendrian cube number}. We will show that the Legendrian cube number distinguishes the Legendrian left hand torus knots with maximal Thurston-Bennequin number and maximal rotation number from the Legendrian left hand torus knots with maximal Thurston-Bennequin number and minimal rotation number.

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。

具有 Horadam 参数的扭曲环面纽结

Twisted torus knots with Horadam parameters

本文研究具有 Horadam 递归参数的扭曲环面纽结，构造推广 Sangyop Lee 工作的纽结族，并建立其与 primitive/primitive 及 primitive/Seifert 扭曲环面纽结列表的联系。

1-桥环面纽结的参数化

Parameterizations of 1-bridge torus knots

亏格为$\le 1$的3-流形中的单桥环面纽结是在Heegaard环面上画出的具有一个桥的纽结。我们给出了两种类型的范式来参数化单桥环面结族，它们类似于2桥结的舒伯特范式和康威范式。对于给定的Schubert规范形，我们给出了算法来确定组件的数量和计算的基本群的补充时，规范形确定一个结。利用其Conway标准形，给出了沿1-桥环面纽结分支的周围3-流形的双分支覆盖的一个描述，并得到了双分支覆盖的第一同调的一个显式公式.

Knot contact homology detects cabled, composite, and torus knots

Knot contact homology is an invariant of knots derived from Legendrian contact homology which has numerous connections to the knot group. We use basic properties of knot groups to prove that knot contact homology detects every torus knot. Further, if the knot contact homology of a knot is isomorphic to that of a cable (respectively composite) knot, then the knot is a cable (respectively composite).

Untwisting 3-strand torus knots

We prove that the signature bound for the topological 4-genus of 3-strand torus knots is sharp, using McCoy's twisting method. We also show that the bound is off by at most 1 for 4-strand and 6-strand torus knots, and improve the upper bound on the asymptotic ratio between the topological 4-genus and the Seifert genus of torus knots from 2/3 to 14/27.

环面纽结横截数的界

Bounds on the Crosscap Number of Torus Knots

对于一个环面纽结K，我们用亏格g（K）和交叉数n（K）来定义交叉数c（K）：c（K）\leq [（g（K）+9）/6]和c（K）\leq [（n（K）+16）/12].（6 n-2，3）环面结表明这些边界是尖锐的。