Norbert A'Campo
We deduce from a rooted tree in the disk a slalom divide and a slalom knot. A slalom knot is either the local link of a simple plane curve singularity of type A_2n, E_6, E_8 or a fibered hyperbolic knot with very special monodromy.
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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.
A three-variable transcendental invariant of planar knotoids via Gauss diagrams
该研究通过高斯图索引函数提出平面类纽结的三变量超越不变量，证明其为一阶Vassiliev不变量，并用于给出同伦平面类纽结Gordian距离的下界。
The stick index of a knot is the least number of line segments required to build the knot in space. We define two analogous 2-dimensional invariants, the planar stick index, which is the least number of line segments in the plane to build a projection, and the spherical stick index, which is the least number of great circle arcs to build a projection on the sphere. We find bounds on these quantities in terms of other knot invariants, and give planar stick and spherical stick constructions for torus knots and for compositions of trefoils. In particular, unlike most knot invariants,we show that …
From annular to toroidal pseudo knots
本文将平面伪纽结理论推广到环形与环面情形，通过提升、O-混合与H-混合表示建立理论联系，并推广加权分解集以构造新不变量。
New Quantum Invariants of Planar Knotoids
该研究将类纽结用于开曲线纽结建模，定义双标架平面类纽结并构造可多项式时间计算的新量子不变量，改进了交叉数≤5的平面类纽结的分类。
From planar to annular to toroidal bracket polynomials for pseudo knots and links
本文将Kauffman括号和Jones型多项式从平面伪链环推广到环面和轮胎面伪链环，并针对其混合链环图表示引入相应多项式，为研究环面和轮胎面伪链环提供了新工具。
We consider compact 3-manifolds M having a submersion h to R in which each generic point inverse is a planar surface. The standard height function on a submanifold of the 3-sphere is a motivating example. To (M, h) we associate a connectivity graph G. For M in the 3-sphere, G is a tree if and only if there is a Fox reimbedding of M which carries horizontal circles to a complete collection of complementary meridian circles. On the other hand, if the connectivity graph of the complement of M is a tree, then there is a level-preserving reimbedding of M so that its complement is a connected sum of…
In 2010, Turaev introduced knotoids as a variation on knots that replaces the embedding of a circle with the embedding of a closed interval with two endpoints. A variety of knot invariants have been extended to knotoids. Here we provide definitions of hyperbolicity for both spherical and planar knotoids. We prove that the product of hyperbolic spherical knotoids is hyperbolic and the volumes add. We also determine the least volume of a rational spherical knotoid and provide various classes of hyperbolic knotoids. We also include tables of hyperbolic volumes for both spherical and planar knotoi…
We introduce a new way to tabulate knots by representing knot diagrams using a pair of planar trees. This pair of trees have their edges labeled by integers, they have no valence 2 vertices, and they have the same number of valence 1 vertices. The number of valence 1 vertices of the trees is called the girth of the knot diagram. The classification problem of knots admitting girth 2 and 3 diagrams is studied. The planar tree pair representations of girth less than 4 for knot diagrams in Rolfsen's table are given.
In this paper we construct some invariants of spatial graphs by disk-summing the constituent knots and show the delta edge-homotopy invariance of them. As an application, we show that there exist infinitely many slice spatial embeddings of a planar graph up to delta edge-homotopy, and there exist infinitely many boundary spatial embeddings of a planar graph up to delta edge-homotopy.
The KnotMosaics Package for SageMath
本文介绍了一个SageMath工具包，用于构建、可视化和分析结马赛克图。该工具包实现了验证马赛克、追踪丝线和组件、计算平面图代码等功能，支持结和链的计算。
It is shown that given any link-manifold, there is an algorithm to decide if the manifold contains an embedded, essential planar surface; if it does, the algorithm will construct one. If a slope on the boundary of the link-manifold is given, there is an algorithm to determine if the slope bounds an embedded punctured-disk; if a meridian slope is given, then it can be determined if a longitude bounds an embedded punctured-disk. The methods use normal surface theory but do not follow the classical approach. Properties of minimal vertex triangulations, layered-triangulations, 0--efficient triangu…
Virtual knot theory is a generalization (discovered by the author in 1996) of knot theory to the study of all oriented Gauss codes. (Classical knot theory is a study of planar Gauss codes.) Graph theory studies non-planar graphs via graphical diagrams with virtual crossings. Virtual knot theory studies non-planar Gauss codes via knot diagrams with virtual crossings. This paper gives basic results and examples (such as non-trivial virtual knots with trivial Jones polynomial), studies fundamental group and quandles of virtual knots, extensions of the bracket and Jones polynomials, quantum link i…
Planar graphs deformations of bordered knot algebras
本文为k=1情形下带边纽结代数上的A-无穷结构提供显式组合模型及其加权形变。
Recent advances in Quantum Topology assign $q$-series to knots in at least three different ways. The $q$-series are given by generalized Nahm sums (i.e., special $q$-hypergeometric sums) and have unknown modular and asymptotic properties. We give an efficient method to compute those $q$-series that come from planar graphs (i.e., reduced Tait graphs of alternating links) and compute several terms of those series for all graphs with at most 8 edges drawing several conclusions. In addition, we give a graph-theory proof of a theorem of Dasbach-Lin which identifies the coefficient of $q^k$ in those…
We show that a small tree-decomposition of a knot diagram induces a small sphere-decomposition of the corresponding knot. This, in turn, implies that the knot admits a small essential planar meridional surface or a small bridge sphere. We use this to give the first examples of knots where any diagram has high tree-width. This answers a question of Burton and of Makowsky and Mariño.
This is a survey article for the forthcoming `A Concise Encyclopedia of Knot Theory.' We focus on the topology of spatial graphs with few vertices and edges, paying particular attention to Brunnian $θ$-graphs.
Planar decomposition of the HOMFLY polynomial for bipartite knots and links
本文提出了一种适用于由反平行锁缠结构成的二部纽结图的全新平面分解方法，将Kauffman括号理论从sl(2)推广到任意sl(N)，从而为HOMFLY多项式的计算提供了与树状演算互补的新途径，并开辟了通向同调技术的新方向。
Knot contact homology as a planar limit of Chern-Simons theory
该研究通过证明HOMFLYPT差模块与结接触同调的关系，揭示了Chern-Simons理论在大N极限下的结构特性。
An ideal triangulation $\mathcal{T}$ of a hyperbolic 3-manifold $M$ with one cusp is non-peripheral if no edge of $\mathcal{T}$ is homotopic to a curve in the boundary torus of $M$. For such a triangulation, the gluing and completeness equations can be solved to recover the hyperbolic structure of $M$. A planar projection of a knot gives four ideal cell decompositions of its complement (minus 2 balls), two of which are ideal triangulations that use 4 (resp., 5) ideal tetrahedra per crossing. Our main result is that these ideal triangulations are non-peripheral for all planar, reduced, alternat…
We show the existence of an infinite collection of hyperbolic knots where each of which has in its exterior meridional essential planar surfaces of arbitrarily large number of boundary components, or, equivalently, that each of these knots has essential tangle decompositions of arbitrarily large number of strings. Moreover, each of these knots has in its exterior meridional essential surfaces of any positive genus and (even) number of boundary components. That is, the compact surfaces that have a meridional essential embedding into a hyperbolic knot exterior have meridional essential embedding…
We reconsider the su(3) link homology theory defined by Khovanov in math.QA/0304375 and generalized by Mackaay and Vaz in math.GT/0603307. With some slight modifications, we describe the theory as a map from the planar algebra of tangles to a planar algebra of (complexes of) `cobordisms with seams' (actually, a `canopolis'), making it local in the sense of Bar-Natan's local su(2) theory of math.GT/0410495. We show that this `seamed cobordism canopolis' decategorifies to give precisely what you'd both hope for and expect: Kuperberg's su(3) spider defined in q-alg/9712003. We conjecture an answe…
Bordered Heegaard Floer modules for satellite operations using planar graphs
本文通过组合构造$(p,1)$-缆线卫星运算的加权$A_\infty$-模，并利用装饰平面图计数给出其$A_\infty$结构关系的组合证明及唯一性结果。
Virtual knots are associated with knot diagrams, which are not obligatory planar. The recently suggested generalization from N=2 to arbitrary N of the Kauffman-Khovanov calculus of cycles in resolved diagrams can be straightforwardly applied to non-planar case. In simple examples we demonstrate that this construction preserves topological invariance -- thus implying the existence of HOMFLY extension of cabled Jones polynomials for virtual knots and links.
A Seifert algorithm for integral homology spheres
本文将经典纽结理论中的 Seifert 算法推广到任意闭积分同调 3-球面，利用 Heegaard 分裂的平面图表示和有向缠结的正则投影，构造出 Seifert 圆及半扭转带，从而证明每个有向链环都是有向嵌入曲面的边界。
Immersed flat ribbon knots
该研究通过将浸入式平面带状纽结链环空间嵌入圆盘图空间求解最小带长，给出反例与猜想，计算了部分小型及无限族链环图的最小带长，还得到了对应最小带长的图的交叉数界。
We construct link invariants using the $D_{2n}$ subfactor planar algebras, and use these to prove new identities relating certain specializations of colored Jones polynomials to specializations of other quantum knot polynomials. These identities can also be explained by coincidences between small modular categories involving the even parts of the $D_{2n}$ planar algebras. We discuss the origins of these coincidences, explaining the role of $SO$ level-rank duality, Kirby-Melvin symmetry, and properties of small Dynkin diagrams. One of these coincidences involves $G_2$ and does not appear to be …
We study collections of planar curves that yield diagrams for all knots. In particular, we show that a very special class called potholder curves carries all knots. This has implications for realizing all knots and links as special types of meanders and braids. We also introduce and apply a method to compare the efficiency of various classes of curves that represent all knots.
In this paper, we show that any topological knot or link in $S^1 \times S^2$ sits on a planar page of an open book decomposition whose monodromy is a product of positive Dehn twists. As a consequence, any knot or link type in $S^1 \times S^2$ has a Legendrian representative having support genus zero. We also show this holds for some knots and links in the lens spaces $L(p,1)$.
Given a crossing in a planar diagram of a link in the three-sphere, we show that the knot Floer homologies of the link and its two resolutions at that crossing are related by an exact triangle. As a consequence, we deduce that for any quasi-alternating knot, the total rank of its knot Floer homology is equal to the determinant of the knot.
Clock Moves and Alexander Polynomial of Plane Graphs
本文通过引入平面图生成树的钟面移动概念，建立了亚历山大多项式的生成树模型，证明了系数序列的单峰性，从而证实了平面奇异纽结的梯形猜想并为Fox交错纽结猜想提供新见解。
Arnold introduced invariants $J^+$, $J^-$ and $St$ for generic planar curves. It is known that both $J^+ /2 + St$ and $J^- /2 + St$ are invariants for generic spherical curves. Applying these invariants to underlying curves of knot diagrams, we can obtain lower bounds for the number of Reidemeister moves for uknotting. $J^- /2 + St$ works well for unmatched RII moves. However, it works only by halves for RI moves. Let $w$ denote the writhe for a knot diagram. We show that $J^- /2 + St \pm w/2$ works well also for RI moves, and demonstrate that it gives a precise estimation for a certain knot d…
We give constructions to realize an odd number, which is representable as sum of two squares, as determinant of an achiral knot, thus proving that these are exactly the numbers occurring as such determinants. Later we study which numbers occur as determinants of prime alternating achiral knots, and obtain a complete result for perfect squares. Using the checkerboard coloring, then an application is given to the number of spanning trees in planar self-dual graphs. Another application are some enumeration results on achiral rational knots. Finally, we describe the leading coefficients of the Ale…
A Structural Approach to Tree Decompositions of Knots and Spatial Graphs
本文研究了结和空间图的树分解，定义了禁止低树宽图示的障碍，并证明其最优性，针对高表示性结（如环面结）提供了新的证明，表明它们不具有低树宽图示。
Besides offering a friendly introduction to knot homologies and quantum curves, the goal of these lectures is to review some of the concrete predictions that follow from the physical interpretation of knot homologies. In particular, this interpretation allows one to pose questions that would not have been asked otherwise, such as, "Is there a direct relation between Khovanov homology and the A-polynomial of a knot?" We will explain that the answer to this question is "yes," and introduce a certain deformation of the planar algebraic curve defined by the zero locus of the A-polynomial. This nov…
A plane curve is a knot diagram in which each crossing is replaced by a 4-valent vertex, and so are dual to a subset of planar quadrangulations. The aim of this paper is to introduce a new tool for sampling diagrams via sampling of plane curves. At present the most efficient method for sampling diagrams is rejection sampling, however that method is inefficient at even modest sizes. We introduce Markov chains that sample from the space of plane curves using local moves based on Reidemeister moves. By then mapping vertices on those curves to crossings we produce random knot diagrams. Combining t…
Persistent Khovanov homology of tangles
本文提出缠结的持续Khovanov同调框架，通过具体函子和平面代数构造，解决现有方法难以捕捉纽结与链环局部拓扑特征的问题，为曲线型数据拓扑分析提供理论基础与实践策略。
From disc patterns in the plane to character varieties of knot groups
本文针对由链环解结隧道产生的无限体积双曲3维流形，提出简洁通用的完整群表示空间构造与计算可视化方法，以支持特征簇中的猜想发现与代数计算。
We provide an upper bound on the Cheeger constant and first eigenvalue of the Laplacian of a finite-volume hyperbolic 3-manifold M, in terms of data from any surgery diagram for M. This has several consequences. We prove that a family of hyperbolic alternating link complements is expanding if and only if they have bounded volume. We also provide examples of hyperbolic 3-manifolds which require 'complicated' surgery diagrams, thereby proving that a recent theorem of Constantino and Thurston is sharp. Along the way, we find a new upper bound on the bridge number of a knot that is not tangle comp…
Incompressible planar surfaces in hyperbolic link exteriors in the 3-sphere
对于每个整数N≥3，构造N分支双曲链的例子，其外部包含一个不可压缩的跨越平面曲面，该曲面在每个边界环面上具有非经向且非整数斜率，从而反驳了Eudave-Muñoz和Ozawa的猜想。
A theta curve is a spatial embedding of the $θ$-graph in the three-sphere, taken up to ambient isotopy. We define the determinant of a theta curve as an integer-valued invariant arising from the first homology of its Klein cover. When a theta curve is simple, containing a constituent unknot, we prove that the determinant of the theta curve is the product of the determinants of the constituent knots. Our proofs are combinatorial, relying on Kirchhoff's Matrix Tree Theorem and spanning tree enumeration results for symmetric, signed, planar graphs.
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