Yi Ni
Let $K$ be a genus $g$ alternating knot with Alexander polynomial $Δ_K(T)=\sum_{i=-g}^ga_iT^i$. We show that if $|a_g|=|a_{g-1}|$, then $K$ is the torus knot $T_{2g+1,\pm2}$. This is a special case of the Fox Trapezoidal Conjecture. The proof uses Ozsváth and Szabó's work on alternating knots.
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In this paper we define Crossing Change Alternating Knots (CCA knots) and their generalization: $k$-CCA knots.
We extend Howie's characterization of alternating knots to give a topological characterization of toroidally alternating knots, which were defined by Adams. We provide necessary and sufficient conditions for a knot to be toroidally alternating. We also give a topological characterization of almost-alternating knots which is different from Ito's recent characterization.
The almost alternating diagrams of the trivial knot
Bankwitz的特点是交替图代表平凡结。一个非交错图称为几乎交错图，如果一个交叉变化使图交错。我们刻画了一个表示平凡纽结的几乎交错图。作为推论，我们确定了一个解结数为1的交错纽结，它具有解结运算可以在其交错图上进行的性质。
This is a short expository article on alternating knots and is to appear in the Concise Encyclopedia of Knot Theory.
Generalizing Howie and Greene's characterization of alternating knots, we give a topological characterization of almost alternating knots.
There is an infinitely generated free subgroup of the smooth knot concordance group with the property that no nontrivial element in this subgroup can be represented by an alternating knot. This subgroup has the further property that every element is represented by a topologically slice knot.
For any alternating knot, it is known that the double branched cover of the $3$-sphere branched over the knot is an $L$-space. We show that the three-fold cyclic branched cover is also an $L$-space for any genus one alternating knot.
A conjecture proposed by J. Tripp in 2002 states that the crossing number of any knot coincides with the canonical genus of its Whitehead double. In the meantime, it has been established that this conjecture is true for a large class of alternating knots including $(2, n)$ torus knots, $2$-bridge knots, algebraic alternating knots, and alternating pretzel knots. In this paper, we prove that the conjecture is not true for any alternating $3$-braid knot which is the connected sum of two torus knots of type $(2, m)$ and $(2, n)$. This results in a new modified conjecture that the crossing number …
Classification of alternating knots with tunnel number one
本文给出了洞数为1的交错纽结及其解纽结洞的完整分类。我们证明，只有这样的结是双桥结和某些Montesinos结。
We show that there are infinitely many pairs of alternating pretzel knots whose Jones polynomials are identical.
We give a complete classification of toroidal Seifert fibered surgeries on alternating knots. Precisely, we show that if an alternating knot admits a toroidal Seifert fibered surgery, then the knot is either the trefoil knot and the surgery slope is zero, or the connected sum of a (2,p)-torus knot and a (2,q)-torus knot and the surgery slope is 2(p+q) with |p|, |q| at least three.
Among non-alternating knots with $n\le 12$ crossings given in \cite{1} Turaev genus is not known for 191 knot. For 154 of them we show that they are almost alternating, so their Turaev genus is 1.
Conjecture $\mathbb{Z}$ is a knot theoretical equivalent form of the Kervaire Conjecture. We say that a knot have property $\mathbb{Z}$ if it satisfies Conjecture $\mathbb{Z}$ for that specific knot. In this work, we show that alternating Montesinos knots with three tangles have property $\mathbb{Z}$. We also show that all the pretzel knots of the form $P(p,q,r)$ (not necessarily alternating) have property $\mathbb{Z}$.
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.
Construction of non-alternating knots
我们研究了Rasmussen不变量$s$在结的sharp运算下的行为，得到了sharp解结数的一个下界.这个界限使我们产生了一个有趣的移动，将任意的结转换为非交替的结。
All exceptional surgeries on alternating knots are integral surgeries
我们表明，所有特殊的手术双曲交替结在3-球面的积分手术。
Lefschetz Fibrations on Knot Traces of Alternating and Extended Alternating Knots
本文基于之前的工作，将构造正允许Lefschetz纤维的方法应用于交替结和扩展交替结的结迹，证明其纤维的亏格等于平面图中的白区域数，并得出正预结结和正扭转结的结迹的纤维亏格结果。
Special Alternating Knots are Band Prime
本文证明特殊交错纽结不能分解为非平凡带和，由此限制其协合关系，并提出和验证特殊交错纽结是带状协合极小的猜想，受相关向下封闭猜想启发。
It was previously shown by the second author that every knot in $S^3$ is ambient isotopic to one component of a two-component, alternating, hyperbolic link. In this paper, we define the alternating volume of a knot $K$ to be the minimum volume of any link $L$ in a natural class of alternating, hyperbolic links such that $K$ is ambient isotopic to a component of $L$. Our main result shows that the alternating volume of a knot is coarsely equivalent to the twist number of a knot.
Let K be a an alternating prime knot in the 3-sphere. We investigate the category of flypes between reduced alternating diagrams for K. As a consequence, we show that any odd prime order action on K is isotopic through maps of pairs to a single flype. This implies that for any odd prime order action on K there is either a reduced alternating periodic diagram or a reduced alternating free periodic diagram. Finally, we deduce that the quotient of an odd periodic alternating knot is also alternating.
Non-triviality of generalized alternating knots
在这篇文章中，我们考虑交替纽结上的一个闭曲面在3-球，并表明，这些是不平行的任何闭曲面不相交的规定。
We give a topological characterisation of alternating knot exteriors based on the presence of special spanning surfaces. This shows that alternating is a topological property of the knot exterior and not just a property of diagrams, answering an old question of Fox. We also give a characterisation of alternating link exteriors which have marked meridians. We then describe a normal surface algorithm which can decide if a knot is prime and alternating given a triangulation of its exterior as input.
The warping degree of an oriented knot diagram is the minimal number of crossing changes which are required to obtain a monotone knot diagram from the diagram. The minimal warping degree of a knot is the minimal value of the warping degree for all oriented minimal diagrams of the knot. In this paper, all prime alternating knots with minimal warping degree two are determined.
We show that, for an alternating knot, the ratio of the diameter of the set of boundary slopes to the crossing number can be arbitrarily large.
A pair of surgeries on a knot is chirally cosmetic if they result in homeomorphic manifolds with opposite orientations. Using recent methods of Ichihara, Ito, and Saito, we show that, except for the (2,5) and (2,7)-torus knots, the genus 2 and 3 alternating odd pretzel knots do not admit any chirally cosmetic surgeries. Further, we show that for a fixed genus, at most finitely many alternating odd pretzel knots admit chirally cosmetic surgeries.
We extend the Neumann's methods and give the explicit formulae for the volume and the Chern-Simons invariant for hyperbolic alternating knot orbifolds.
In 1999, Kauffman-Harary conjectured that every non-trivial Fox $p$-coloring of a reduced, alternating knot diagram with prime determinant $p$ is heterogeneous. Ten years later this conjecture was proved by W. Mattman and P. Solis. Mathew Williamson generalized this conjecture to alternating virtual knots and proved it for certain families of virtual knots. In the present note, we use the methods of W. Mattman and P. Solis to give an affirmative answer to the Kauffman-Harary conjecture for alternating virtual knots.
We describe an algorithm to find ribbon disks for alternating knots, and the results of a computer implementation of this algorithm. The algorithm is underlain by a slice link obstruction coming from Donaldson's diagonalisation theorem. It successfully finds ribbon disks for slice two-bridge knots and for the connected sum of any alternating knot with its reverse mirror, as well as for 662,903 prime alternating knots of 21 or fewer crossings. We also identify some examples of ribbon alternating knots for which the algorithm fails to find ribbon disks, though a related search identifies all suc…
This paper is devoted to prove the existence of $q$-periodic alternating projections of prime alternating $q$-periodic knots. The main tool is the Menasco-Thistlethwaite's Flyping theorem. Let $K$ be an oriented prime alternating knot that is $q$-periodic with $q\geq 3$, i.e. $K$ admits a symmetry that is a rotation of order $q$. Then $K$ has an alternating $q$-periodic projection. As applications, we obtain the crossing number of a $q$ -periodic alternating knot with $q\geq 3$ is a multiple of $q$ and we give an elementary proof that the knot $12_{a634} $ is not 3-periodic; this proof does no…
We specify the computational complexity of crosscap numbers of alternating knots by introducing an automatic computation. For an alternating knot $K$, let $\cal{E}$ be the number of edges of its diagram. Then there exists a code such that the complexity of this computation of the crosscap number of $K$ is estimated by $O(\cal{E}^3)$.
For any integer n > 2, the n-fold cyclic branched cover M of an alternating prime knot K in the 3-sphere determines K, meaning that if K is a knot in the 3-sphere that is not equivalent to K then its n-fold cyclic branched cover cannot be homeomorphic to M.
Using exhaustive techniques and results from Lackenby and many others, we compute the tunnel number of all 1655 alternating 11 and 12 crossing knots and of 881 non-alternating 11 and 12 crossing knots. We also find all 5525 Montesinos knots with 14 crossings or fewer.
Obstructing Unknotting Number Two For Some Alternating Knots
该研究利用Ozsváth-Szabó的d不变量与Brendan Owens的定理，算出194个13交叉以内交错纽结的解结数，为纽结理论研究提供了具体数据支撑。
We prove that for any zero α of the Alexander polynomial of a two-bridge knot, -3 < Re(α) < 6. Furthermore, for a large class of two-bridge knots we prove -1<Re(α).
Every knot can be embedded in the union of finitely many half planes with a common boundary line in such a way that the portion of the knot in each half plane is a properly embedded arc. The minimal number of such half planes is called the arc index of the knot. We have identified all prime knots with arc index up to 11. We also proved that the crossing number is an upperbound of arc index for non-alternating knots. As a result the arc index is determined for prime knots up to twelve crossings.
We show that if the branched double cover of an alternating link arises as $p/q \in \mathbb{Q} \setminus \mathbb{Z}$ surgery on a knot in $S^3$, then this is exhibited by a rational tangle replacement in an alternating diagram.
For any given integer $r \geq 1$ and a quasitoric braid $β_r=(σ_r^{-ε} σ_{r-1}^ε...$ $ σ_{1}^{(-1)^{r}ε})^3$ with $ε=\pm 1$, we prove that the maximum degree in $z$ of the HOMFLYPT polynomial $P_{W_2(\hatβ_r)}(v,z)$ of the doubled link $W_2(\hatβ_r)$ of the closure $\hatβ_r$ is equal to $6r-1$. As an application, we give a family $\mathcal K^3$ of alternating knots, including $(2,n)$ torus knots, 2-bridge knots and alternating pretzel knots as its subfamilies, such that the minimal crossing number of any alternating knot in $\mathcal K^3$ coincides with the canonical genus of its Whitehead dou…
An alternating distance is a link invariant that measures how far away a link is from alternating. We study several alternating distances and demonstrate that there exist families of links for which the difference between certain alternating distances is arbitrarily large. We also show that two alternating distances, the alternation number and the alternating genus, are not comparable.
Weakly generalised alternating knots are knots with an alternating projection onto a closed surface in a compact irreducible 3-manifold, and they share many hyperbolic geometric properties with usual alternating knots. For example, usual alternating knots have volume bounded above and below by the twist number of the alternating diagram due to Lackenby. Howie and Purcell showed that a similar lower bound holds for weakly generalised alternating knots. In this paper, we show that a generalisation of the upper volume bound does not hold, by producing a family of weakly generalised alternating kn…
Large alternating Montesinos knots do not admit purely cosmetic surgeries
本文通过分析扭转数和长度的下界条件，证明大型交错 Montesinos 纽结以及若干相关纽结族不容许纯装饰 Dehn 手术，从而为 3-球面中非平凡纽结的纯装饰手术猜想提供了新的支持证据。
We show that the difference between the genus and the stable topological 4-genus of alternating knots is either zero or at least 1/3.
We prove that if an alternating knot has unknotting number one, then there exists an unknotting crossing in any alternating diagram. This is done by showing that the obstruction to unknotting number one developed by Greene in his work on alternating 3-braid knots is sufficient to identify all unknotting number one alternating knots. As a consequence, we also get a converse to the Montesinos trick: an alternating knot has unknotting number one if its branched double cover arises as half-integer surgery on a knot in $S^3$. We also reprove a characterisation of almost-alternating diagrams of the …
We show for an alternating knot the minimal boundary slope of an essential spanning surface is given by the signature plus twice the minimum degree of the Jones polynomial and the maximal boundary slope of an essential spanning surface is given by the signature plus twice the maximum degree of the Jones polynomial. For alternating Montesinos knots, these are the minimal and maximal boundary slopes.
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…
In this article, we give a list of minimal grid diagrams of the 12 crossing prime alternating knots. This is a continuation of the work in https://doi.org/10.1142/S0218216520500765
We study the geometry of hyperbolic knots that admit alternating projections on embedded surfaces in closed 3-manifolds. We show that, under mild hypothesis, their cusp area admits two sided bounds in terms of the twist number of the alternating projection and the genus of the projection surface. As a result, we derive diagrammatic estimates of slope lengths and give applications to Dehn surgery. These generalize results of Lackenby and Purcell about alternating knots in the 3-sphere. Using a result of Kalfagianni and Purcell, we point out that alternating knots on surfaces of higher genus, ca…
A volume-ish theorem for the Jones polynomial of alternating knots
体积猜想声称一个纽结的双曲体积是由有色琼斯多项式决定的。本文的目的是用琼斯多项式而不是有色琼斯多项式来证明交替结点的体积定理：琼斯多项式的体积与某些系数之和的比值从上到下都由常数限定。此外，我们给出了实验数据的增长的双曲线体积和琼斯多项式的系数的关系，交替和非交替结。
In this paper we are interested in symmetries of alternating knots, more precisely in those related to achirality. We call the following statement Tait's Conjecture on alternating -achiral knots: Let K be an alternating -achiral knot. Then there exists a minimal projection Π of K in S^2 \subset S^3 and an involution ϕ:S^3\toS^3 such that: 1) ϕ reverses the orientation of $S^3$; 2) ϕ(S^2) = S^2; 3) ϕ (Π) = Π; 4) ϕ has two fixed points on Π and hence reverses the orientation of K. The purpose of this paper is to prove this statement.
Ito-Takimura recently defined a splice-unknotting number $u^-(D)$ for knot diagrams. They proved that this number provides an upper bound for the crosscap number of any prime knot, asking whether equality holds in the alternating case. We answer their question in the affirmative. (Ito has independently proven the same result.) As an application, we compute the crosscap numbers of all prime alternating knots through at least 13 crossings, using Gauss codes.
Quasi-alternating surgeries on asymmetric L-space knots in the census
本文针对SnapPy普查中的9个非对称L-空间纽结，借助Montesinos技巧验证了此前通过计算机得到的每个纽结恰有两种拟交错手术的结论。
Quasi-alternating Montesinos links
本文的目的是检测新的类准交替链接。准交替链接是交替链接的自然推广。它们的结Floer和Khovanov同调特别容易计算。由于结弗洛尔同调检测结的属以及结是否纤维化，提供了解结数和切片属的界限，准交替链接的表征成为一个有趣的开放问题。我们表明，存在类的非交替Montesinos链接，这是准交替。
We investigate the coefficients of the highest and lowest terms (also called the head and the tail) of the colored Jones polynomial and show that they stabilize for alternating links and for adequate links. To do this we apply techniques from skein theory.
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…
A note on alternating knots in handlebodies
本文研究手柄体中交替结的最小交叉数和旋转数，并推广Jones多项式到该设置中。
We show that a special alternating knot with sufficiently large number (more than $63$) of twist regions has no chirally cosmetic surgeries, a pair of Dehn surgeries producing orientation-reversingly homeomorphic $3$-manifolds. In the course of proof, we provide the optimal upper bounds of the primitive finite type invariants of degree 2 and 3 that solve Willerton's conjecture.
Let $D$ be a diagram of an alternating knot with unknotting number one. The branched double cover of $S^3$ branched over $D$ is an L-space obtained by half integral surgery on a knot $K_D$. We denote the set of all such knots $K_D$ by $\mathcal D$. We characterize when $K_D\in \mathcal D$ is a torus knot, a satellite knot or a hyperbolic knot. In a different direction, we show that for a given $n>0$, there are only finitely many L-space knots in $\mathcal D$ with genus less than $n$.
Khovanov Homology and the Twist Number of Alternating Knots
X.S. Lin和O.达斯巴赫证明了交替纽结的琼斯多项式的第二个和倒数第二个系数的绝对值之和等于纽结的捻数。本文利用Khovanov同调给出了其结果的一个新的证明。证明是通过归纳的交叉数使用长的准确序列在Khovanov同调对应的考夫曼括号绞关系。
Quasi-alternating surgeries
本文研究了准交替手术的性质，证明所有SnapPy名录中的L-space结除了两个例外外都允许准交替手术，并展示了非对称L-space结的斜率特征。
We construct an infinite family of hyperbolic, homologically thin knots that are not quasi-alternating. To establish the latter, we argue that the branched double-cover of each knot in the family does not bound a negative definite 4-manifold with trivial first homology and bounded second betti number. This fact depends in turn on information from the correction terms in Heegaard Floer homology, which we establish by way of a relationship to, and calculation of, the Turaev torsion.
This article is devoted to the study of prime alternating +achiral knots. In the case of arborescent knots, we prove in +AAA Visibility Theorem 5.1, that the symmetry is visible on a certain projection (not necessarily minimal) and that it is realised by a homeomorphism of order 4. In the general case (arborescent or not), if the prime alternating knot has no minimal projection on which +achirality is visible, we prove that the order of +achirality is necessarily equal to 4.
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