Finite order invariants for $(n,2)$-torus knots and the curve $Y^2=X^3+X^2$
Svetlana Tyurina, Alexander Varchenko
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。
We describe the algebra of finite order invariants on the set of all $(n,2)$-torus knots.
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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.
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.
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 $pqs$.
If a knot is a nontrivial connected sum of positive torus knots, then it is not concordant to an L-space knot.
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。
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.
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…
We connect Dedekind sums and Alexander polynomials of torus knots.
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。
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.
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…
In the present note, we will show that there are infinitely many composite twisted 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.
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））。
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.
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.
We show that the distortion of the (2,q)-torus knot is not bounded linearly from below.
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.
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.
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。
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.
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.
We show that certain negatively twisted torus knots admit Dehn surgeries yielding 3-manifolds with non left-orderable fundamental groups.
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.
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.
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。
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.
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.
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。
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 $D(n,n+1)$.
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.
We show that a torus knot which is not 2-bridge has a unique irreducible bridge splitting of positive genus.
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。
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.
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.
Let $r$ be an odd integer, $r\ge3$. Then the petal number of the torus knot of type $(r,r+2)$ is equal to $2r+3$.
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.
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.
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.
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。
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.
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$m>0$。
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。
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.
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.
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。
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。
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.
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.
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。
Knot concordance plays a crucial role in the low dimensional topology. We propose a very elementary techniques which allows one to construct a lot of sliceness obstructions for knots in the full torus. Our approach deals with group theoretical techniques; it is completely combinatorial, and the groups are very easy to deal with.
A non-torus positive 2-bridge knot does not admit chirally cosmetic surgeries
该研究证明了非环面的正2桥结不接受镜像化妆品手术，通过Dehn手术得到反向定向同胚的3流形。
We determine the set of all genus g bridge numbers of many iterated torus knots, listing these numbers in a sequence called the bridge spectrum. In addition, we prove a structural lemma about the decomposition of a strongly irreducible bridge surface induced by cutting along a collection of essential surfaces.
We prove that the topological locally flat slice genus of large torus knots takes up less than three quarters of the ordinary genus. As an application, we derive the best possible linear estimate of the topological slice genus for torus knots with non-maximal signature invariant.
We define a new kind of Gauss diagrams to describe knots in the solid torus with projections in the annulus. We see that it provides an efficient tool for showing that a knot diagram can be fully recovered from its decorated Gauss diagram, and we use it to establish a characterization of the decorated Gauss diagrams of closed braids.
(1,1)-knots via the mapping class group of the twice punctured torus
利用二次穿孔环面MCG（T，2）的映射类群，给出了（1，1）-纽结的代数表示.证明了透镜空间L（p，q）中的每个（1，1）-纽结都可以用MCG（T，2）的某个秩为2的自由子群的元素与一个仅依赖于周围空间的标准元的合成来表示.作为一个值得注意的例子，我们得到了这种类型的所有环面结和所有双桥结的表示。此外，我们给出了（k，ck+2）型环面纽结的循环分支覆盖的基本群的明确的循环表示。
It is known that any surface knot can be transformed to an unknotted surface knot or a surface knot which has a diagram with no triple points by a finite number of 1-handle additions. The minimum number of such 1-handles is called the unknotting number or the triple point cancelling number, respectively. In this paper, we give upper bounds and lower bounds of unknotting numbers and triple point cancelling numbers of torus-covering knots, which are surface knots in the form of coverings over the standard torus $T$. Upper bounds are given by using $m$-charts on $T$ presenting torus-covering knot…
Difference equation of the colored Jones polynomial for torus knot
证明了环面纽结T_{s,t}的N色Jones多项式满足二阶差分方程，对于T_{2,2m+1}情形，该差分方程可化为一阶差分方程.我们表明，A-多项式的环面结可以从这个差分方程。还构造了T_{2,2m+1}的有色Jones多项式的q-超几何型表达式。
On the colored Jones polynomials of certain cable of the torus knots
本文研究了一类特殊纽结的色琼斯多项式在单位根上的渐近性态。我们证明了体积猜想所预言的极限为零。
We show that for a torus knot the SL(2;C) Chern-Simons invariants and the SL(2;C) twisted Reidemeister torsions appear in an asymptotic expansion of the colored Jones polynomial. This suggests a generalization of the volume conjecture that relates the asymptotic behavior of the colored Jones polynomial of a knot to the volume of the knot complement.
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