Robert E. Tuzun, Adam S. Sikora
Extending upon our previous work, we verify the Jones Unknot Conjecture for all knots up to $24$ crossings. We describe the method of our approach and analyze the growth of the computational complexity of its different components.
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The Jones unknot conjecture states that the Jones polynomial distinguishes the unknot from nontrivial knots. We prove it for knots up to 23 crossings.
We address the question: Does there exist a non-trivial knot with a trivial Jones polynomial? To find such a knot, it is almost certainly sufficient to find a non-trivial braid on four strands in the kernel of the Burau representation. I will describe a computer algorithm to search for such a braid.
We proved by computer enumeration that the Jones polynomial distinguishes the unknot for knots up to 22 crossings. Following an approach of Yamada, we generated knot diagrams by inserting algebraic tangles into Conway polyhedra, computed their Jones polynomials by a divide-and-conquer method, and tested those with trivial Jones polynomials for unknottedness with the computer program SnapPy. We employed numerous novel strategies for reducing the computation time per knot diagram and the number of knot diagrams to be considered. That made computations up to 21 crossings possible on a single proc…
It is still unknown whether there is a nontrivial knot with Jones polynomial equal to that of the unknot. Tanaka shows that if an amphichiral knot is a symmetric union of the unknot with one twist region, then its Jones polynomial is trivial. Hence, he proposes that if any of these knots were nontrivial, a nontrivial knot with trivial Jones polynomial would exist. We first show such a knot is always trivial and hence cannot be used to answer the above question. We then generalize the argument to symmetric unions of any knots and show that if a symmetric union of a knot $J$ with one twist regio…
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 show that there is a knot satisfying the property that for each minimal crossing number diagram of the knot and each single crossing of the diagram, changing the crossing results in a diagram for a knot whose unknotting number is at least that of the original knot, thus giving a counterexample to the Bernhard-Jablan Conjecture.
The unknotting number, hard unknot diagrams, and reinforcement learning
本文利用强化学习智能体寻找纽结图的最短解结交叉变换序列，确定了大量纽结的解结数，证明了连通和图示中单次解结变换可产生素纽结，并构建了260万困难解结图数据集。
New presentations of a link and a virtual link are introduced and algebraic systems on links and virtual links are constructed respectively. Based on the algebraic systems, Reduction Crossing Algorithms for them are proposed which are used to reduce the number of crossings in a link and virtual link. For known unknots, one can transform them into a trivial knot in a polynomial time by applying corresponding algorithm. As special consequences, Goeritz's unknot and Thistlethwaite's unknot are unknotted. Moreover, an infinite family of knots $K_{G_{2k,2l}}$ are unknotted in $O(n^2)$ time where $n…
Closed 4-braids and the Jones unknot conjecture
针对Jones纽结猜想中4股辫子情形的未解问题，研究发现一族以拧数为参数、可能对应非平凡纽结的双变量HOMFLY-PT多项式（其Jones多项式平凡），探究其性质并提出检验方法与多股推广思路。
We present three "hard" diagrams of the unknot. They require (at least) three extra crossings before they can be simplified to the trivial unknot diagram via Reidemeister moves in $\mathbb{S}^2$. Both examples are constructed by applying previously proposed methods. The proof of their hardness uses significant computational resources. We also determine that no small "standard" example of a hard unknot diagram requires more than one extra crossing for Reidemeister moves in $\mathbb{S}^2$.
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。
Computation of unknotting numbers: which knot breaks the Bernhard-Jablan Conjecture
本文计算了至多13交叉的素纽结的2525个解结数，利用Heegaard Floer障碍和生成树模型确定下界，并给出交叉变更构造上界，最终以13n3370作为反例推翻Bernhard-Jablan猜想。
Knot Probabilities in Random Diagrams
该研究提出随机纽结图模型，通过统计10交叉点及以下的纽结图分类计算精确概率，发现平凡纽结占比高（10交叉点图约78%且源于树状图特性，且纽结概率对数与排名对数近似符合Zipf定律。
Tangle Equations, the Jones conjecture, slopes of surfaces in tangle complements, and q-deformed rationals
本文研究2-缠结方程组的解的唯一性，证明带标架有理解至多一个，并指出Jones解结猜想蕴含一般唯一性；同时引入Kauffman括号比并猜想其与曲面斜率及q-有理数的关系，还关联了两个著名猜想。
Using an involved study of the Jones polynomial, we determine, as our main result, the crossing numbers of (prime) amphicheiral knots. As further applications, we show that several classes of links, including semiadequate links and Whitehead doubles of semiadequate knots, have non-trivial Jones polynomial. We also prove that there are infinitely many positive knots with no positive minimal crossing diagrams. Some relations to the twist number of a link, Mahler measure and the hyperbolic volume are given, for example explicit upper bounds on the volume for Montesinos and 3-braid links in terms …
Hard unknots are often easy from a different perspective
针对难解未结图，提出ReAPR方法交替进行Pass移动简化与几何重嵌入，成功简化所有已知难解未结及约260万个新示例，总CPU耗时不足30秒。
In the present paper a criteria for a rectangular diagram to admit a simplification is given in terms of Legendrian knots. It is shown that there are two types of simplifications which are mutually independent in a sense. A new proof of the monotonic simplification theorem for the unknot is given. It is shown that a minimal rectangular diagram maximizes the Thurston--Bennequin number for the corresponding Legendrian links. Jones' conjecture about the invariance of the algebraic number of intersections of a minimal braid representing a fixed link type is proved.
Jones-Wenzl projectors and the Khovanov homotopy of the infinite twist
本文将Jones-Wenzl投影算子提升至Khovanov谱，通过映射序列实现该提升，并给出3股谱投影算子上的多项式作用以计算三着色Khovanov谱，证明了Lobb-Orson-Schütz猜想，同时推翻了Lawson-Lipshitz-Sarkar猜想。
It has long been known that the quadratic term in the degree of the colored Jones polynomial of a knot is bounded above in terms of the crossing number of the knot. We show that this bound is sharp if and only if the knot is adequate. As an application of our result we determine the crossing numbers of broad families of non-adequate prime satellite knots. More specifically, we exhibit minimal crossing number diagrams for untwisted Whitehead doubles of zero-writhe adequate knots. This allows us to determine the crossing number of untwisted Whitehead doubles of any amphicheiral adequate knot, in…
Some petal diagrams of the unknot are hard
本文通过穷举计算机搜索证明，平凡结的花瓣图在最多7个花瓣时可单调简化，但9个花瓣时存在108个困难图，其中三个在对称性下本质不同，揭示了花瓣图与矩形图在简化性质上的差异。
Untwisting information from Heegaard Floer homology
本文研究解结数的推广——解扭数，证明Montesinos技巧不能用于阻碍低解扭数，但通过Heegaard Floer校正项给出解扭数为1的条件，并利用tau和s不变量区分不同p-解扭数。
We give examples of knots with some unusual properties of the crossing number of positive diagrams or strand number of positive braid representations. In particular we show that positive braid knots may not have positive minimal (strand number) braid representations, giving a counterpart to results of Franks-Williams and Murasugi. Other examples answer questions of Cromwell on homogeneous and (partially) of Adams on almost alternating knots. We give a counterexample to, and a corrected version of a theorem of Jones on the Alexander polynomial of 4-braid knots. We also give an example of a knot…
We show that two Dehn surgeries on a knot $K$ never yield manifolds that are homeomorphic as oriented manifolds if $V_K''(1)\neq 0$ or $V_K'''(1)\neq 0$. As an application, we verify the cosmetic surgery conjecture for all knots with no more than $11$ crossings except for three $10$-crossing knots and five $11$-crossing knots. We also compute the finite type invariant of order $3$ for two-bridge knots and Whitehead doubles, from which we prove several nonexistence results of purely cosmetic surgery.
On the Cosmetic Crossing Conjecture for Special Alternating Links
该研究证明含所有特殊交替纽结的一类链环无保同痕的非平凡交叉改变，结合Lidman、Moore的相关结果与Scharlemann、Thompson的证明工具推进了cosmetic crossing猜想的研究。
Khovanov homology is a categorification of the Jones polynomial, so it may be seen as a kind of quantum invariant of knots and links. Although polynomial quantum invariants are deeply involved with Vassiliev (aka. finite type) invariants, the relation remains unclear in case of Khovanov homology. Aiming at it, in this paper, we discuss a categorified version of Vassiliev skein relation on Khovanov homology. More precisely, we will show that the "genus-one" operation gives rise to a crossing change on Khovanov complexes. Invariance under Reidemeister moves turns out, and it enables us to extend…
We give a brief historical overview of the Tait conjectures, made 120 years ago in the course of his pioneering work in tabulating the simplest knots, and solved a century later using the Jones polynomial. We announce the solution, again based on a substantial study of the Jones polynomial, of one (possibly his last remaining?) problem of Tait, with the construction of amphicheiral knots of almost all odd crossing numbers. An application to the non-triviality problem for the Jones polynomial is also outlined.
There are infinitely many pretzel links with the same Alexander polynomial (actually with trivial Alexander polynomial). By contrast, in this note we revisit the Jones polynomial of pretzel links and prove that, given a natural number S, there is only a finite number of pretzel links whose Jones polynomials have span S. More concretely, we provide an algorithm useful for deciding whether or not a given knot is pretzel. As an application we identify all the pretzel knots up to nine crossings, proving in particular that $8_{12}$ is the first non-pretzel knot.
We examine the structure and dimensionality of the Jones polynomial using manifold learning techniques. Our data set consists of more than 10 million knots up to 17 crossings and two other special families up to 2001 crossings. We introduce and describe a method for using filtrations to analyze infinite data sets where representative sampling is impossible or impractical, an essential requirement for working with knots and the data from knot invariants. In particular, this method provides a new approach for analyzing knot invariants using Principal Component Analysis. Using this approach on th…
The cosmetic crossing conjecture (also known as the "nugatory crossing conjecture") asserts that the only crossing changes that preserve the oriented isotopy class of a knot in the 3-sphere are nugatory. We use the Dehn surgery characterization of the unknot to prove this conjecture for knots in integer homology spheres whose branched double covers are L-spaces satisfying a homological condition. This includes as a special case all alternating and quasi-alternating knots with square-free determinant. As an application, we prove the cosmetic crossing conjecture holds for all knots with at most …
The Slope Conjecture proposed by Garoufalidis asserts that the degree of the colored Jones polynomial determines a boundary slope, and its refinement, the Strong Slope Conjecture proposed by Kalfagianni and Tran asserts that the linear term in the degree determines the topology of an essential surface that satisfies the Slope Conjecture. Under certain hypotheses, we show that Mazur doubles of knots satisfy the Strong Slope Conjecture if the original knot does. Consequently, any knot obtained by a finite sequence of cabling, untwisted w--generalized Whitehead doublings with w > 0, connected sum…
The virtual skein relation for the Jones polynomial of the virtual link diagram was introduced by N. Kamada, S. Nakabo, and S. Satoh. H. A. Dye, L. H. Kauffman, and Y. Miyazawa introduced multivariable polynomial, an invariant of virtual links, which is a refinement of Jones polynomial. In this paper, we give a skein relation for the multivariable polynomials among positive, negative, and virtual crossings with some restrictions. We apply this relation to study some properties of virtual links obtained by replacing a real crossing by a virtual crossing.
The paper introduces Slope Conjecture which relates the degree of the Jones polynomial of a knot and its parallels with the slopes of incompressible surfaces in the knot complement. More precisely, we introduce two knot invariants, the Jones slopes (a finite set of rational numbers) and the Jones period (a natural number) of a knot in 3-space. We formulate a number of conjectures for these invariants and verify them by explicit computations for the class of alternating knots, torus knots, the knots with at most 9 crossings, and the $(-2,3,n)$ pretzel knots.
On an alternative stratification of knots
本文提出按纽结可被放置的最小格点尺寸进行分层，计算不同格点内平凡纽结及高交叉数纽结比例，并验证了理论估计。
A knot in the three-sphere is doubly slice if it is the cross-section of an unknotted two-sphere in the four-sphere. For low-crossing knots, the most complete work to date gives a classification of doubly slice knots through 9 crossings. We extend that work through 12 crossings, resolving all but four cases among the 2,977 prime knots in that range. The techniques involved in this analysis include considerations of the Alexander module and signature functions as well as calculations of the twisted Alexander polynomials for higher-order branched covers. We give explicit illustrations of the dou…
Let D be a link diagram with n crossings, s_A and s_B its extreme states and |s_AD| (resp. |s_BD|) the number of simple closed curves that appear when smoothing D according to s_A (resp. s_B). We give a general formula for the sum |s_AD|+|s_BD| for a k-almost alternating diagram D, for any k, characterising this sum as the number of faces in an appropriate triangulation of an appropriate surface with boundary. When D is dealternator connected, the triangulation is especially simple, yielding |s_AD|+|s_BD|=n+2-2k. This gives a simple geometric proof of the upper bound of the span of the Jones p…
A Simple Characterization of Adequate Links
本文利用Jones多项式广度与Turaev亏数等数值不变量，给出了adequate链环的简单刻画，并应用于阻止拟交错性及确定交叉数下界。
Champanerkar and Kofman introduced an interesting way to construct new examples of quasi-alternating links from existing ones. Actually, they proved that replacing a quasi-alternating crossing c in a quasi-alternating link by a rational tangle of same type yields a new quasi-alternating link. This construction has been extended to alternating algebraic tangles and applied to characterize all quasi-alternating Montesinos links. In this paper, we extend this technique to any alternating tangle of same type as c. As an application, we give new examples of quasi-alternating knots of 13 and 14 cros…
Jones Polynomials and their Zeros for a Family of Knots and Links
本文通过计算关联图的Tutte多项式并作变量替换，给出了一族任意多交叉数交错纽结与链环的琼斯多项式，规避了指数级计算复杂度，并分析了交叉数趋于无穷时多项式零点的聚点集合。
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