Region crossing change on surfaces
Jiawei Cheng, Zhiyun Cheng, Jinwen Xu, Jieyao Zheng
本文研究（修正的）区域交叉变换在更高亏格曲面上的行为，并给出了其可解性条件。
区域交叉变换是链环图上的局部操作。在$S^2$上的区域交叉变换行为已被充分理解。本文研究（修正的）区域交叉变换在更高亏格曲面上的行为。
Region crossing change is a local operation on link diagrams. The behavior of region crossing change on $S^2$ is well understood. In this paper, we study the behavior of (modified) region crossing change on higher genus surfaces.
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We introduce a local move on a link diagram named a region freeze crossing change which is close to a region crossing change, but not the same. We study similarity and difference between region crossing change and region freeze crossing change.
In this paper we define Crossing Change Alternating Knots (CCA knots) and their generalization: $k$-CCA knots.
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Region crossing change on nonorientable surfaces
本文研究不可定向曲面上链环图的区域交叉变换，并给出了相应的完整分类。
When is arc crossing change an unknotting operation?
本文扩展弧交叉变换至链环图，探讨其作为解结操作的条件，并证明交错纽结图中任意两交叉点均可进行该变换。
In this paper, we prove that region crossing change on a link diagram is an unknotting operation if and only if the link is proper. A description of the behavior of region crossing change on link diagrams is given. Furthermore we also discuss the relation between region crossing change and the Arf invariant of proper links.
A crossing in a knot is nugatory if changing the crossing does not change the knot type. Using an invariant of certain types of closed 3-braid diagrams, we show that if a closed 3-braid contains a nugatory crossing then its braid index is one or two. This proves a special case of a conjecture on nugatory crossings due to Xiao-Song Lin.
Region crossing change is a local transformation on a knot or link diagram. We show that a region crossing change on a knot diagram is an unknotting operation, and we define the region unknotting numbers for a knot diagram and a knot.
A region crossing change is a local transformation on spatial graph diagrams switching the over/under relations at all the crossings on the boundary of a region. In this paper, we show that a spatial graph of a planar graph is unknottable by region crossing changes if and only if the spatial graph is non-Eulerian or is Eulerian and proper.
We extend the crossing-change maps between grid complexes, defined by Ozsváth-Szabó-Stipsicz, to filtered grid complexes and give a combinatorial formulation of the Alishahi-Eftekhary $\mathfrak{l}(K)$ knot invariant.
We analyze all monodromies of genus one fibered knots that possess clean or once-unclean arcs, and use this to determine all manifolds containing genus one fibered knots with generalized crossing changes resulting in another genus one fibered knot, and classify all such generalized crossing changes between two genus one, fibered knots.
We show that if K is a satellite knot which admits a generalized cosmetic crossing change of order q with |q| \geq 6, then K admits a pattern knot with a generalized cosmetic crossing change of the same order. As a consequence of this, we find that any prime satellite knot which admits a pattern knot that is fibered cannot admit a generalized cosmetic crossing changes of order q with |q| \geq 6. We also show that if there is any knot admitting a generalized cosmetic crossing change of order q with |q| \geq 6, then there must be such a knot which is hyperbolic.
Crossing number of curves on surfaces
该研究聚焦曲面上简单闭曲线系统的交叉数最小化问题，确定了亏格2曲面上最多12条曲线的最小交叉数，且证明最小化系统在曲面同胚与曲线同痕下唯一。
In a recent work of Ayaka Shimizu$^{[5]}$, she defined an operation named region crossing change on link diagrams, and showed that region crossing change is an unknotting operation for knot diagrams. In this paper, we prove that region crossing change on a 2-component link diagram is an unknotting operation if and only if the linking number of the diagram is even. Besides, we define an incidence matrix of a link diagram via its signed planar graph and its dual graph. By studying the relation between region crossing change and incidence matrix, we prove that a signed planar graph represents an …
We compute lower bounds on the virtual crossing number and minimal surface genus of virtual knot diagrams from the arrow polynomial. In particular, we focus on several interesting examples.
Crossings of knot diagrams can be divided into classes (tribes) compatible with Reidemeister moves. Tribes can be considered as localization of the notion of weak chord index introduced by M. Xu. In the article we describe tribes of crossings for tangles in a fixed surface and show that crossings are differentiated by their component, order and homotopy types. As a consequence, we conclude that there are no nontrivial indices on diagrams of classical knots.
Region crossing change on origami and link
本文将原本定义在纽结投影上的Region Select游戏拓展至折纸折痕图案展开研究，并定义探讨了与区域交叉变换相关的新解环数。
A symmetric union of two knots is a classical construction in knot theory which generalizes connected sum, introduced by Kinoshita and Terasaka in the 1950s. We study this construction for the purpose of finding an infinite family of hyperbolic non-fibered three-bridge knots of constant determinant which satisfy the well-known cosmetic crossing conjecture. This conjecture asserts that the only crossing changes which preserve the isotopy type of a knot are nugatory.
We characterize cutting arcs on fiber surfaces that produce new fiber surfaces, and the changes in monodromy resulting from such cuts. As a corollary, we characterize band surgeries between fibered links and introduce an operation called Generalized Hopf banding. We further characterize generalized crossing changes between fibered links, and the resulting changes in monodromy.
On the crossing number of knots and links on surface in 3-manifolds
利用基本群的秩，给出三维流形中双侧曲面上纽结和链环投影的交叉数下界。
An Alexander Polynomial Obstruction to Cosmetic Crossing Changes
本文推广先前研究，提出一个 Alexander 多项式条件，用于阻碍具有 $L$-空间分支二重覆盖的纽结发生装饰性交叉变换，并据此证明了五参数无限族 Pretzel 纽结的装饰性交叉猜想。
We use technology from sutured manifold theory and the theory of Heegaard splittings to relate genus reducing crossing changes on knots in S^3 to twists on surfaces arising in circular Heegaard splittings for knot complements. In a separate paper, currently in preparation, we prove that these circular Heegaard splittings may be searched for algorithmically, and together our results imply that an algorithm to detect when two hyperbolic or fibered knots of different genus are related by a crossing change would follow from an algorithm to determine whether two compact oriented surfaces in S^3 are…
We prove the nugatory crossing conjecture for fibered knots. We also show that if a knot $K$ is $n$-adjacent to a fibered knot $K'$, for some $n>1$, then either the genus of $K$ is larger than that of $K'$ or $K$ is isotopic to $K'$.
Gordian complex of knots was defined by Hirasawa and Uchida as the simplicial complex whose vertices are knot isotopy classes in $\mathbb{S}^3$. Later Horiuchi and Ohyama defined Gordian complex of virtual knots using $v$-move and forbidden moves. In this paper we discuss Gordian complex of knots by region crossing change and Gordian complex of virtual knots by arc shift move. Arc shift move is a local move in the virtual knot diagram which results in reversing orientation locally between two consecutive crossings. We show the existence of an arbitrarily high dimensional simplex in both the Go…
Boolean algebra on region crossing change and an inequality of the region unknotting number
该研究运用布尔代数分析纽结的区域交叉变换，将任意交叉数为 c 的纽结的区域解结数上界由 (c+2)/2 改进为 (c+1)/2。
In this article, we investigate the problem of counting totally geodesic surfaces in the complement of hyperbolic knots with at most 9 crossings. Adapting previous counting techniques of boundary slope and intersection, we establish uniqueness of a totally geodesic surface for the knots $7_4$ and $9_{35}$. Extending an obstruction to the existence of totally geodesic surfaces due to Calegari, we show that there is no totally geodesic surface in the complement of 47 knots.
We prove that the property of admitting no cosmetic crossing changes is preserved under the operation of forming certain satellites of winding number zero. We also define strongly cosmetic crossing changes and we discuss their behavior under the operation of inserting full twists in the strings of closed braids.
In this paper, we discuss degree 0 crossing change on Khovanov homology in terms of cobordisms. Namely, using Bar-Natan's formalism of Khovanov homology, we introduce a sum of cobordisms that yields a morphism on complexes of two diagrams of crossing change, which we call the "genus-one morphism." It is proved that the morphism is invariant under the moves of double points in tangle diagrams. As a consequence, in the spirit of Vassiliev theory, taking iterated mapping cones, we obtain an invariant for singular tangles that extending sl(2) tangle homology; examples include Lee homology, Bar-Nat…
Distinguishing surface-links with 4-charts with 2 crossings and 8 black vertices
本文使用图表研究曲面链环，通过quandle着色方法证明了具有2个交叉点和8个黑色顶点的4-图表第二类中的图表表示不同的曲面链环。
Let K' be a knot that admits no cosmetic crossing changes and let C be a non-trivial, prime, non-cable knot. Then any knot that is a satellite of C with winding number zero and pattern K' admits no cosmetic crossing changes. As a consequence we prove the nugatory crossing conjecture for Whitehead doubles of prime, non-cable knots.
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…
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。
Crossing Numbers of Knots on Closed Surfaces
本文研究了结在闭曲面上的交叉数与隧道数、Heegaard缺陷之间的关系，揭示了Heegaard缺陷如何度量曲面能吸收的隧道复杂度而不产生交叉。
Any knot in $S^3$ may be reduced to a slice knot by crossing changes. Indeed, this slice knot can be taken to be the unknot. In this paper we study the question of when the same holds for knots in homology spheres. We show that a knot in a homology sphere is nullhomotopic in a smooth homology ball if and only if that knot is smoothly concordant to a knot which is homotopic to a smoothly slice knot. As a consequence, we prove that the equivalence relation on knots in homology spheres given by cobounding immersed annuli in a homology cobordism is generated by concordance in homology cobordisms t…
Region crossing change on planar trivalent graphs
本文以平面三价图为对象，考察区域穿越变换所产生的作用。
Let $D$ be a reduced alternating diagram of a non-split link $L$ and $\tilde{L}$ be the link whose diagram is obtained from $D$ by a crossing change. If $\tilde{L}$ is alternating, then $c(\tilde{L})\leq c(L)-2$. In this paper we explore when $c(\tilde{L})=c(L)-2$ holds and obtain a simple sufficient and necessary condition in terms of plane graphs corresponding to $L$. This result is obtained via analyzing the behavior of the Tutte polynomial of the signed plane graph corresponding to $\tilde{L}$.
In this paper, we discuss the crossing change operation along exchangeable double curves of a surface-knot diagram. We show that under certain condition, a finite sequence of Roseman moves preserves the property of those exchangeable double curves. As an application for this result, we also define a numerical invariant for a set of surface-knots called $du$-exchangeable set.
Genus non-increasing totally positive unknotting number
该研究定义了亏格非递增全正解结数这一纽结不变量，证明亏格为1的纽结的该不变量可任意大，拓展了纽结交叉改变与亏格关系的相关研究。
We develop a word mechanism applied in knot and link diagrams for the illustration of a diagrammatic property. We also give a necessary condition for determining incompressible and pairwise incompressible surfaces, that are embedded in knot or link complements. Finally, we give a finiteness theorem and an upper bound on the Euler characteristic of such surfaces.
Meier and Zupan proved that an orientable surface $\mathcal{K}$ in $S^4$ admits a tri-plane diagram with zero crossings if and only if $\mathcal{K}$ is unknotted, so that the crossing number of $\mathcal{K}$ is zero. We determine the minimal crossing numbers of nonorientable unknotted surfaces in $S^4$, proving that $c(\mathcal{P}^{n,m}) = \max\{1,|n-m|\}$, where $\mathcal{P}^{n,m}$ denotes the connected sum of $n$ unknotted projective planes with normal Euler number $+2$ and $m$ unknotted projective planes with normal Euler number $-2$. In addition, we convert Yoshikawa's table of knotted sur…
The crossing number of polynomial curve systems
本文研究了标准曲面上多项式规模曲线系统的交叉数，以亏格为参数，确定了高精度的计算方法。
In this paper we investigate the unlinking numbers of 10-crossing links. We make use of various link invariants and explore their behaviour when crossings are changed. The methods we describe have been used previously to compute unlinking numbers of links with crossing number at most 9. Ultimately, we find the unlinking numbers of all but 2 of the 287 prime, non-split links with crossing number 10.
Crossing matrix and a polynomial invariant of braid systems up to Hurwitz equivalence
研究辫子的交叉矩阵，引入一个在Hurwitz等价下不变的辫子系统多项式不变量，并应用于曲面辫子和曲面链接，定义了一个指示辫子系统之间是否需要欧拉融合或分裂的不变量。
We introduce a new polynomial invariant of virtual knots and links and use this invariant to compute a lower bound on the virtual crossing number and the minimal surface genus.
We show that a finite numerical boundary slope of an essential surface in the exterior of a Montesinos knot is bounded above and below in terms of the numbers of positive/negative crossings of a specific minimal diagram of the knot.
A (weak chord) index is a function on the crossings of knot diagrams such that: 1) the index of a crossing does not change under Reidemeister moves; 2) crossings which can be paired by a second Reidemeister move have the same index. We show that one can omit the second condition in the case of the universal index. As a consequence, we get the following principle of indistinguishability for classical knots: crossings of the same sign in a classical knot diagram can not be distinguished by any inherent property.
In the present paper, we construct an invariant for virtual knots in the thickened sphere with g handles; this invariant is a Laurent polynomial in 2g+3 variables. To this end, we use a modification of the Wirtinger presentation of the knot group and the concept of parity introduced by V.O.Manturov. The section 4 of the paper is devoted to an enhancement of the invariant (construction of the invariant module) by using the parity hierarchy concept suggested by V.O.Manturov. Namely, we discriminate between odd crossings and two types of even crossings; the latter two types depend on whether an e…
We show that the following unlinking strategy does not always yield an optimal sequence of crossing changes: first split the link with the minimal number of crossing changes, and then unknot the resulting components.
Reachability under arc crossing changes
本文改进了Cericola关于纽结图可通过弧交叉变换为升式未纽结图的结果，明确了R₁-约化经典纽结影子上交叉数大于3的图的可达性关系，通过标记高斯字的局部五出现条件刻画源图。
Signature, slicing foams, and crossing changes of Klein graphs
本文推广Klein图的签名，建立其与切片亏格和解结数的下界关系，并改进已知解结数界限。
In links with two components there are three different types of crossings: self-crossings in the first component, self crossings in the second component, and crossings between components. In this paper we examine the minimum number of crossing changes needed to unlink without changing the crossings between components. We restrict our attention to unlinking two component links with linking number zero and both components unknotted. We provide data for links with no more than ten crossings and general results about asymmetry of unlinking between components.
Given a surface with boundary and some points on its boundary, a polygon diagram is a way to connect those points as vertices of non-overlapping polygons on the surface. Such polygon diagrams represent non-crossing permutations on a surface with any genus and number of boundary components. If only bigons are allowed, then it becomes an arc diagram. The count of arc diagrams is known to have a rich structure. We show that the count of polygon diagrams exhibits the same interesting behaviours, in particular it is almost polynomial in the number of points on the boundary components, and the leadi…
The linear minimal 4-chart with three crossings
本文研究了通过4图描述嵌入4维空间中的表面，证明任何具有三个交叉的线性最小4图与2-twist spun trefoil knot的图通过删除自由边和环等价。
Crossing number inequalities for curves on surfaces
本文证明曲面上 $m$ 条同伦不同闭曲线的交叉数至少以 $(m \log m)^2$ 增长，据此回答了关于多重图非同伦边画法交叉数的问题，并在所有可定向曲面上推广得到最优增长率的界。
We introduce a new Legendrian isotopy invariant for any closed orientable Legendrian surface $L$ embedded in a closed contact $5$-manifold $(M, ξ)$ which admits an "admissable" open book $(B, f)$ (supporting $ξ$) for $L$. We show that to any such $L$ and a fixed page $X$, one can assign an integer $M\mathcal{P}_{X}(L)$, called "Relative Maximal Page Crossing Number of $L$ with respect to $X$", which is invariant under Legendrian isotopies of $L$. We also show that one can extend this to a page-free invariant, i.e., one can assign an integer $M\mathcal{P}_{(B,f)}(L)$, called "Absolute Maximal P…
Modular systoles are extremal for the crossing number
本文研究同余曲面上收缩圈的交叉数，证明其相交增长速率在同拓扑曲面且相同基数的所有曲线集合中达到最优最小。
We provide an algorithm to determine whether a link L admits a crossing change that turns it into a split link, under some fairly mild hypotheses on L. The algorithm also provides a complete list of all such crossing changes. It can therefore also determine whether the unlinking number of L is 1.
A diagonal surface in a link exterior M is a properly embedded, incompressible, boundary incompressible surface which furthermore has the same number of boundary components and same slope on each component of the boundary of M. We derive a formula for the boundary slope of a diagonal surface in the exterior of a 2-bridge link which is analogous to the formula for the boundary slope of a 2-bridge knot found by Hatcher and Thurston. Using this formula we show that the diameter of a 2-bridge link, that is, the difference between the smallest and largest finite slopes of diagonal surfaces, is equa…
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