Jae Choon Cha, Charles Livingston
This paper, to be regularly updated, lists those prime knots with the fewest possible number of crossings for which values of basic knot invariants, such as the unknotting number or the smooth 4-genus, are unknown. This list is being developed in conjunction with "KnotInfo" (www.indiana.edu/~knotinfo), a web-based table of knot invariants.
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This note corrects errors in Hatcher and Oertel's table of boundary slopes of Montesinos knots which have projections with 10 or fewer crossings.
A table of genus two handlebody-knots with seven crossings
研究具有七个交叉的亏格为二的柄体纽结，通过枚举扩展石井等人的表格。
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…
Yoshikawa made a table of knotted surfaces in R^4 with ch-index 10 or less. This remarkable table is the first to enumerate knotted surfaces analogous to the classical prime knot table. A broken sheet diagram of a surface-link is a generic projection of the surface in R^3 with crossing information along its singular set. The minimal number of triple points among all broken sheet diagrams representing a given surface-knot is its triple point number. This paper compiles the known triple point numbers of the surface-links represented in Yoshikawa's table and calculates or provides bounds on the t…
This work presents formulas for the Kauffman bracket and Jones polynomials of 3-bridge knots using the structure of Chebyshev knots and their billiard table diagrams. In particular, these give far fewer terms than in the Skein relation expansion. The subject is introduced by considering the easier case of 2-bridge knots, where some geometric interpretation is provided, as well, via combinatorial tiling problems.
We present an enhanced prime decomposition theorem for knots that gives the isotopy classes of composite knots that can be constructed from a given list of prime factors (allowing for the mirroring and orientation reversing for each factor). Underlying the theorem is an algebraic construction that also allows for the computation of the intrinsic symmetries (invertibility, chirality, etc.) of a composite knot from those of the prime factors. We then use this construction to give a table of composite knots through 12 crossings that can be constructed from prime factors through 9 crossings. This …
To study embeddings of tangles in knots, we use quandle cocycle invariants. Computations are carried out for the tables of knots and tangles, to investigate which tangles may or may not embed in knots in the tables.
A polynomial pair invariant of alternating knots and links
本文提出一种基于交错纽结最小图示中两个棋盘平面图的整数多项式对不变量WRP，证明其良定义，并能区分表中所有交叉数不超过10的纽结及其镜像，强度可与经典不变量如三变量考夫曼括号相比。
This is a facsimile of the circa 1990 unpublished manuscript with the same title. All the original text, figures and tables are included; although text has been reset in \TeX, the original hand-drawn figures have been redrawn digitally, and the parameter $k$ in the original table of lens spaces has been replaced with the originally intended $λ$. And, of course, this abstract has been added.
We compose the table of knots in the thickened torus T x I having diagrams with at most 4 crossings. The knots are constructed by the three-step process. First we list regular graphs of degree 4 with at most 4 vertices, then for each graph we enumerate all corresponding knot projections, and after that we construct the corresponding minimal diagrams. Several known and new tricks made it possible to keep the process within reasonable limits and offer a rigorous theoretical proof of the completeness of the table. For proving that all knots are different we use a generalized version of the Kauffm…
Jablan and Radović originally defined two invariants called the Meander number and OGC number of knots for certain classes of knots. We generalize these definitions to all knots and name the straight number and contained straight number of a knot, respectively, and prove they are well defined. We answer two questions and prove a generalization of a conjecture of Jablan and Radović. We also give some relations to crossing number and petal number. Then we compute the straight numbers for all the knots in the standard knot table and present some interesting questions and the complete table of kno…
Extensive rewrite. Tables and proofs have been reformatted and/or rewritten for clarity.
There are many studies about twisted Alexander invariants for knots and links, but calculations of twisted Alexander invariants for spatial graphs, handlebody-knots, and surface-links have not been demonstrated well. In this paper, we give some remarks to calculate the twisted Alexander ideals for spatial graphs, handlebody-knots and surface-links, and observe their behaviors. For spatial graphs, we calculate the invariants of Suzuki's theta-curves and show that the invariants are nontrivial for Suzuki's theta-curves whose Alexander ideals are trivial. For handlebody-knots, we give a remark on…
We present a practical algorithm to determine the minimal genus of non-orientable spanning surfaces for 2-bridge knots, called the crosscap numbers. We will exhibit a table of crosscap numbers of 2-bridge knots up to 12crossings (all 362 of them).
Minimum braids are a complete invariant of knots and links. This paper defines minimum braids, describes how they can be generated, presents tables for knots up to ten crossings and oriented links up to nine crossings, and uses minimum braids to study graph trees, amphicheirality, unknotting numbers, and periodic tables.
We use the Chebyshev knot diagram model of Koseleff and Pecker in order to introduce a random knot diagram model by assigning the crossings to be positive or negative uniformly at random. We give a formula for the probability of choosing a knot at random among all knots with bridge index at most 2. Restricted to this class, we define internal and external reduction moves that decrease the number of crossings of the diagram. We make calculations based on our formula, showing the numerics in graphs and providing evidence for our conjecture that the probability of any knot appearing in this model…
We present a class of knots associated with labelled generic immersions of intervals into the plane and compute their Gordian numbers and 4-dimensional invariants. At least 10% of the knots in Rolfsen's table belong to this class of knots. We call them track knots. They are contained in the class of quasipositive knots. In this connection, we classify quasipositive knots and strongly quasipositive knots up to 10 crossings.
Let $K$ be a tunnel number two knot. Then, by considering the $(g, b)$-decompositions, $K$ is one of (3, 0)-, (2, 1)-, (1, 2)- or (0, 3)-knots. In the present paper, we analyze the connected sum summands of composite tunnel number two knots and give a complete table of those summands from the point of view of $(g, b)$-decompositions.
Symmetric diagrams for all strongly invertible knots up to 10 crossings
该研究提供10交叉以下强可逆纽结的对称图表格，指出其横截图与对称并图的相似性并讨论相关开放问题，为强可逆纽结研究提供基础参考。
The list of knots with up to 10 crossings is commonly referred to as the Rolfsen Table. This paper presents a way to generate the Rolfsen table in a simple, clear, and reproducible manner. The methods we use are similar to those used by J. Hoste, M. Thistlethwaite, and J. Weeks in [1]. The difference between our methods comes from the fact that [1] uses a more complicated algorithm to be able to find all the knots with up to 17 crossings, while our approach demonstrates a simpler way to find the knots up to 10 crossings. We do this by generating all planar knot diagrams with up to 10 crossings…
In this paper we define a set of numerical criteria for a handlebody link to be irreducible. It provides an effective, easy-to-implement method to determine the irreducibility of handlebody links; particularly, it recognizes the irreducibility of all handlebody knots in the Ishii-Kishimoto-Moriuchi-Suzuki knot table and most handlebody links in the Bellettini-Paolini-Paolini-Wang link table.
State-sum invariants for knotted curves and surfaces using quandle cohomology were introduced by Laurel Langford and the authors in math.GT/9903135 In this paper we present methods to compute the invariants and sample computations. Computer calculations of cohomological dimensions for some quandles are presented. For classical knots, Burau representations together with Maple programs are used to evaluate the invariants for knot table. For knotted surfaces in 4-space, movie methods and surface braid theory are used. Relations between the invariants and symmetries of knots are discussed.
The average genus of a 2-bridge knot is asymptotically linear
本文利用台球桌模型研究2-桥结，证明了交叉数为c的2-桥结平均亏格渐近趋于c/4+1/12，从理论上验证了结亏格随交叉数线性增长的实验现象。
For any given number of crossings $c$, there exists a formula to determine the number of 2-bridge knots of $c$ crossings, and indeed it is a simple matter to actually construct presentations of these knots. However, the determination of whether a given (prime) knot is a 2-bridge knot remains a nontrivial exercise, and we have no procedure to determine bridge numbers more generally. Herein, we identify the 2-bridge knots within the Hoste--Thistlethwaite--Weeks tables of prime knots of up to 16 crossings by an exhaustive search of a larger set of 2-bridge knots. As the unknot is the only knot wi…
In a previous work, the first and third authors studied a random knot model for all two-bridge knots using billiard table diagrams. Here we present a closed formula for the distribution of the crossing numbers of such random knots. We also show that the probability of any given knot appearing in this model decays to zero at an exponential rate as the length of the billiard table goes to infinity. This confirms a conjecture from the previous work.
We compute the smooth 4-genera of the prime knots with 12 crossings whose values, as reported on the KnotInfo website, were unknown. This completes the calculation of the smooth 4-genus for all prime knots with 12 or fewer crossings.
We publish a table of primitive finite-type invariants of order less than or equal to six, for knots of ten or fewer crossings. We note certain mod-2 congruences, one of which leads to a chirality criterion in the Alexander polynomial. We state a computational result on mod-2 finite-type invariants of 2-strand string links.
We introduce the notion of a $G$-family of quandles which is an algebraic system whose axioms are motivated by handlebody-knot theory, and use it to construct invariants for handlebody-knots. Our invariant can detect the chiralities of some handlebody-knots including unknown ones.
We construct quantum $\mathcal{U}_q(\mathfrak{sl}_{\,2})$ type invariants for handlebody-knots in the 3-sphere $S^3$. A handlebody-knot is an embedding of a handlebody in a 3-manifold. These invariants are linear sums of Yokota's invariants for colored spatial graphs which are defined by using the Kauffman bracket. We give a table of calculations of our invariants for genus 2 handlebody-knots up to six crossings. We also show our invariants are identified with special cases of the Witten-Reshetikhin-Turaev invariants.
We will strengthen the known upper and lower bounds on the delta-crossing number of knots in therms of the triple-crossing number. The latter bound turns out to be strong enough to obtain (unknown values of) triple-crossing numbers for a few knots. We also prove that we can always find at least one tangle from the set of four tangles, in any triple-crossing projections of any non-trivial knot or non-split link. In the last section, we enumerate and generate tables of minimal delta-diagrams for all prime knots up to the delta-crossing number equal to four. We also give a concise survey about kn…
Tabulating Knot Mosaics: Crossing Number 10 or Less
本研究通过算法编程方法计算了交叉数10或更少的所有素纽结的马赛克数和瓷砖数，并建立了包含马赛克表和用户创作识别工具的在线存储库。
Strongly positive amphicheiral knots with doubly symmetric diagrams
本文分类至多16个交叉的素强正 amphicheiral 纽结，刻画其双重对称或近乎双重对称图，并发现14a19470为表中首个非slice实例。
This paper provides the complete table of prime knot projections with their mirror images, without redundancy, up to eight double points systematically thorough a finite procedure by flypes. In this paper, we show how to tabulate the knot projections up to eight double points by listing tangles with at most four double points by an approach with respect to rational tangles of J. H. Conway. In other words, for a given prime knot projection of an alternating knot, we show how to enumerate possible projections of the alternating knot. Also to tabulate knot projections up to ambient isotopy, we in…
The notion of a pseudoknot is defined as an equivalence class of knot diagrams that may be missing some crossing information. We provide here a topological invariant schema for pseudoknots and their relatives, 4-valent rigid vertex spatial graphs and singular knots, that is obtained by replacing unknown crossings or vertices by rational tangles.
Equivariant crossing numbers for two-bridge knots
本文为二桥结定义了等变交叉数 $c_2(K)$，提供了 $c_t(K)$ 的上界，并给出计算算法，通过计算机计算至14个交叉数，展示了20个至10个交叉数的结例，其对称性可在不增加交叉数下改善。
The concordance genus of a knot K is the minimum three-genus among all knots concordant to K. For prime knots of 10 or fewer crossings there have been three knots for which the concordance genus was unknown. Those three cases are now resolved. Two of the cases are settled using invariants of Levine's algebraic concordance group. The last case depends on the use of twisted Alexander polynomials, viewed as Casson-Gordon invariants.
We consider an analogue of well-known Casson knot invariant for knotoids. We start with a direct analogue of the classical construction which gives two different integer-valued knotoid invariants and then focus on its homology extension. Value of the extension is a formal sum of subgroups of the first homology group $H_1(Σ)$ where $Σ$ is an oriented surface with (maybe) non-empty boundary in which knotoid diagrams lie. To make the extension informative for spherical knotoids it is sufficient to transform an initial knotoid diagram in $S^2$ into a knotoid diagram in the annulus by removing smal…
On the transient number of a knot
本文定义纽结瞬变数tr(K)为使纽结同伦化为平凡纽结需附加的最小简单弧数，给出其基于双分支覆盖第一同调群秩的下界，证明t(K)=1时该群循环，能计算表中纽结瞬变数且存在瞬变数任意大的纽结。
New Upper Bounds for Stick Numbers
本研究利用带纽结类型保持移动的模拟退火方法寻找低杆数多边形纽结，改进了10交叉内素纽结的杆数上界，首次给出13交叉内所有纽结的杆数界表，并确定了19种未知杆数纽结的精确值。
In this paper we study embeddings of oriented connected closed surfaces in $\mathbb S^3$. We define a complete invariant, the fundamental span, for such embeddings, generalizing the notion of the peripheral system of a knot group. From the fundamental span, several computable invariants are derived and employed to study handlebody knots, bi-knotted surfaces, and chirality of knots. These invariants are capable to distinguish inequivalent handlebody knots and bi-knotted surfaces with homeomorphic complements. Particularly, we obtain an alternative proof of the inequivalence of Ishii et al.'s ha…
Tabulation of knots up to five triple-crossings and moves between oriented diagrams
本文枚举了三交叉数不超过五的所有素纽结极小图表，推导了连接同一有向纽结三交叉图的有向移动极小生成集，并提出三交叉数下界与亚历山大多项式宽度相关的猜想。
Arborescent knots are the ones which can be represented in terms of double fat graphs or equivalently as tree Feynman diagrams. This is the class of knots for which the present knowledge is enough for lifting topological description to the level of effective analytical formulas. The paper describes the origin and structure of the new tables of colored knot polynomials, which will be posted at the dedicated site. Even if formal expressions are known in terms of modular transformation matrices, the computation in finite time requires additional ideas. We use the "family" approach, and apply it t…
In this paper we use artificial neural networks to predict and help compute the values of certain knot invariants. In particular, we show that neural networks are able to predict when a knot is quasipositive with a high degree of accuracy. Given a knot with unknown quasipositivity we use these predictions to identify braid representatives that are likely to be quasipositive, which we then subject to further testing to verify. Using these techniques we identify 84 new quasipositive 11 and 12-crossing knots. Furthermore, we show that neural networks are also able to predict and help compute the …
Integer inequalities between knot invariants, skein tree depth and delta-crossing numbers
本文研究了结不变量、skein树深度和delta交叉数之间的整数不等式，通过理论和计算结果证明了新的上界，并给出了新的下界，同时提供了未确定的结和链的skein树深度表。
Every second flat Reidemeister move of knot projections can be decomposed into two types thorough an inverse or direct self-tangency modification, respectively called strong or weak, when orientations of the knot projections are arbitrarily provided. Further, we introduce the notions of strong and weak (1, 2) homotopies; we define that two knot projections are strongly (resp. weakly) (1, 2) homotopic if and only if two knot projections are related by a finite sequence of first and strong (resp. weak) second flat Reidemeister moves. This paper gives a new necessary and sufficient condition that…
In this paper we present a systematic method to generate prime knot and prime link minimal triple-point projections, and then classify all classical prime knots and prime links with triple-crossing number at most four. We also extend the table of known knots and links with triple-crossing number equal to five. By introducing a new type of diagrammatic move, we reduce the number of generating moves on triple-crossing diagrams, and derive a minimal generating set of moves connecting triple-crossing diagrams of the same knot.
We call a knot $K$ a complete Alexander neighbor if every possible Alexander polynomial is realized by a knot one crossing change away from $K$. It is unknown whether there exists a complete Alexander neighbor with nontrivial Alexander polynomial. We eliminate infinite families of knots with nontrivial Alexander polynomial from having this property and discuss possible strategies for unresolved cases. Additionally, we use a condition on determinants of knots one crossing change away from unknotting number one knots to improve KnotInfo's unknotting number data on 11 and 12 crossing knots. Licko…
The twisting number of a ribbon knot is bounded below by its doubly slice genus
该研究证明带状纽结的扭转数以双重切片亏格为下界，构造出扭转数任意高的带状纽结解决Tanaka的问题，并确定了部分12交叉纽结的未知双重切片亏格。
The knots $8_1$, $8_2$, $8_3$, $8_5$, $8_6$, $8_7$, $8_8$, $8_{10}$, $8_{11}$, $8_{12}$, $8_{13}$, $8_{14}$, $8_{15}$, $9_7$, $9_{16}$, $9_{20}$, $9_{26}$, $9_{28}$, $9_{32}$, and $9_{33}$ all have superbridge index equal to 4. This follows from new upper bounds on superbridge index not coming from the stick number and increases the number of knots from the Rolfsen table for which superbridge index is known from 29 to 49. Appendix A gives the current state of knowledge of superbridge index for prime knots through 10 crossings.
We compute the maximal Thurston-Bennequin number for a Legendrian two-bridge knot or oriented two-bridge link in standard contact R^3, by showing that the upper bound given by the Kauffman polynomial is sharp. As an application, we present a table of maximal Thurston-Bennequin numbers for prime knots with nine or fewer crossings.
We construct many examples of non-slice knots in 3-space that cannot be distinguished from slice knots by previously known invariants. Using Whitney towers in place of embedded disks, we define a geometric filtration of the 3-dimensional topological knot concordance group. The bottom part of the filtration exhibits all classical concordance invariants, including the Casson-Gordon invariants. As a first step, we construct an infinite sequence of new obstructions that vanish on slice knots. These take values in the L-theory of skew fields associated to certain {\em universal} groups. Finally, we…
Experimental data from Dunfield et al using random grid diagrams suggests that the genus of a knot grows linearly with respect to the crossing number. Using billiard table diagrams of Chebyshev knots developed by Koseleff and Pecker and a random model of 2-bridge knots via these diagrams developed by the author with Krishnan and then with Even-Zohar and Krishnan, we introduce a further-truncated model of all 2-bridge knots of a given crossing number, almost all counted twice. We present a convenient way to count Seifert circles in this model and use this to compute a lower bound for the averag…
Pseudodiagrams are knot or link diagrams where some of the crossing information is missing. Pseudoknots are equivalence classes of pseudodiagrams, where equivalence is generated by a natural set of Reidemeister moves. In this paper, we introduce a Gauss-diagrammatic theory for pseudoknots which gives rise to the notion of a virtual pseudoknot. We provide new, easily computable isotopy and homotopy invariants for classical and virtual pseudodiagrams. We also give tables of unknotting numbers for homotopically trivial pseudoknots and homotopy classes of homotopically nontrivial pseudoknots. Sinc…
For a knot K in S^3, let T(K) be the characteristic toric sub-orbifold of the orbifold (S^3,K) as defined by Bonahon and Siebenmann. If K has unknotting number one, we show that an unknotting arc for K can always be found which is disjoint from T(K), unless either K is an EM-knot (of Eudave-Munoz) or (S^3,K) contains an EM-tangle after cutting along T(K). As a consequence, we describe exactly which large algebraic knots (ie algebraic in the sense of Conway and containing an essential Conway sphere) have unknotting number one and give a practical procedure for deciding this (as well as determin…
We use Heegaard Floer homology to obtain bounds on unknotting numbers. This is a generalisation of Ozsvath and Szabo's obstruction to unknotting number one. We determine the unknotting numbers of 9_10, 9_13, 9_35, 9_38, 10_53, 10_101 and 10_120; this completes the table of unknotting numbers for prime knots with crossing number nine or less. Our obstruction uses a refined version of Montesinos' theorem which gives a Dehn surgery description of the branched double cover of a knot.
We use recently introduced Rasmussen invariant to find knots that are topologically locally-flatly slice but not smoothly slice. We note that this invariant can be used to give a combinatorial proof of the slice-Bennequin inequality. Finally, we compute the Rasmussen invariant for quasipositive knots and show that most of our examples of non-slice knots are not quasipositive and, to the best of our knowledge, were previously unknown.
Berge in [1] defined doubly primitive knots, which yield lens spaces by Dehn surgery. At the same paper he listed the knots into several types. In this paper we will prove the list is complete when $τ>1$. The invariant $τ$ is a quantity with regard to lens space surgery, which is defined in this paper. Furthermore at the same time we will also prove that Table~6 in [8] is complete as Poincaré homology sphere surgery when $τ>1$.
There is a higher dimensional analogue of the perturbative Chern-Simons theory in the sense that a similar perturbative series as in 3-dimension, which is computed via configuration space integral, yields an invariant of higher dimensional knots (Bott-Cattaneo-Rossi invariant), which is constructed by Bott for degree 2 and by Cattaneo-Rossi for higher degrees. However, its feature is yet unknown. In this paper we restrict the study to long ribbon n-knots and characterize the Bott-Cattaneo-Rossi invariant as a finite type invariant of long ribbon n-knots in [HKS]. As a consequence, we obtain a …
Mosaics for immersed surface-links
本文利用奇异标记图将镶嵌系统推广至浸入曲面链环，并定义其镶嵌数，为量子纽结系统研究提供新方向。
We define the {\it Wirtinger number} of a link, an invariant closely related to the meridional rank. The Wirtinger number is the minimum number of generators of the fundamental group of the link complement over all meridional presentations in which every relation is an iterated Wirtinger relation arising in a diagram. We prove that the Wirtinger number of a link equals its bridge number. This equality can be viewed as establishing a weak version of Cappell and Shaneson's Meridional Rank Conjecture, and suggests a new approach to this conjecture. Our result also leads to a combinatorial techniq…
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