Boundary slope diameter and crossing number of 2-bridge knots
Thomas W. Mattman, Gabriel Maybrun, Kristin Robinson
我们证明，对于2桥结，直径，D，边界斜坡的集合是两倍的交叉数，c。这部分验证了一个猜想，即对于S^3中的所有纽结，D至多为2c。
We prove that for 2-bridge knots, the diameter, D, of the set of boundary slopes is twice the crossing number, c. This constitutes partial verification of a conjecture that, for all knots in S^3, D is at most 2c.
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In this paper we define Crossing Change Alternating Knots (CCA knots) and their generalization: $k$-CCA knots.
We show that for genus one knots the Alexander polynomial and the homology of the double cover branching over the knot provide obstructions to cosmetic crossings. As an application we prove the nugatory crossing conjecture for the negatively twisted, positive Whitehead doubles of all knots. We also verify the conjecture for several families of pretzel knots and all genus one knots with up to 10 crossings.
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Connected sum and crossing numbers of flat virtual knots
研究平坦虚拟纽结连通和对图与基点的依赖性，证明复合纽结的极小交叉图必为连通和图，且交叉数在连通和下具有超加性。
It is known that the arc index of alternating knots is the minimal crossing number plus two and the arc index of prime nonalternating knots is less than or equal to the minimal crossing number. We study some cases when the arc index is strictly less than the minimal crossing number. We also give minimal grid diagrams of some prime nonalternating knots with 13 crossings and 14 crossings whose arc index is the minimal crossing number minus one.
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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.
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.
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The 2-bridge knots of up to 16 crossings
对于任何给定的交叉数$c$，存在一个公式来确定$c$交叉的2桥结的数目，并且实际上构造这些结的表示是一件简单的事情。然而，确定一个给定的（素）结是否是一个2桥结仍然是一个不平凡的练习，我们没有程序来确定更一般的桥数。在这里，我们确定了2桥结内的主机-Thistlethwaite-周表的素数结多达16个路口的一个更大的一组2桥结的详尽搜索。由于非结是唯一桥数为1的结，因此剩余结的桥数的下界为3。
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…
Given a knot in the 3-sphere, the non-orientable 4-genus or 4-dimensional crosscap number of a knot is the minimal first Betti number of non-orientable surfaces, smoothly and properly embedded in the 4-ball, with boundary the knot. In this paper, we calculate the non-orientable 4-genus of knots with crossing number 10.
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.
In this paper, we determine the average genus of all the $2$-bridge knots with a given crossing number. As a consequence, we obtain the oblique asymptote of this value as the crossing number grows.
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We find explicit maximal rank Coxeter quotients for the knot groups of 595,515 out of the 1,701,936 knots through 16 crossings. We thus calculate the bridge numbers and verify Cappell and Shaneson's Meridional Rank Conjecture for these knots. In addition, we provide a computational tool for establishing the conjecture for knots beyond 16 crossings whose meridional ranks can be detected via finite Coxeter quotients.
A triple crossing is a crossing in a projection of a knot or link that has three strands of the knot passing straight through it. A triple crossing projection is a projection such that all of the crossings are triple crossings. We prove that every knot and link has a triple crossing projection and then investigate c_3(K), which is the minimum number of triple crossings in a projection of K. We obtain upper and lower bounds on c_3(K) in terms of the traditional crossing number and show that both are realized. We also relate triple crossing number to the span of the bracket polynomial and use th…
As a continuation of the previous works to tabulate the prime knots up to arc index 11, we provide the list of prime knots with arc index 12 up to 16 crossings and their minimal grid diagrams. There are 19,513 prime knots of arc index 12 up to 16 crossings.
An increasing sequence of integers is said to be universal for knots if every knot has a reduced regular projection on the sphere such that the number of edges of each complementary face of the projection comes from the given sequence. Adams, Shinjo, and Tanaka have, in a work, shown that (2,4,5) and (3,4,n) (where n is a positive integer greater than 4), among others, are universal. In a forthcoming paper, Adams introduces the notion of a multi-crossing projection of a knot. An n-crossing projection} is a projection of a knot in which each crossing has n strands, rather than 2 strands as in a…
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Lower bounds on boundary slope diameters for Montesinos knots
本文给出了Montesinos纽结的边界坡集直径的两个下界。一个是用纽结的最小交叉数来描述的，另一个是与具有最大/最小边界斜率的本质曲面的欧拉示性数有关的。
We prove the cosmetic crossing conjecture for genus one knots with non-trivial Alexander polynomial. We also prove the conjecture for genus one knots with trivial Alexander polynomial, under some additional assumptions.
Boundary slopes and the numbers of positive/negative crossings for Montesinos knots
我们表明，一个有限的数值边界斜率的本质表面在外部的Montesinos结是有界的上下方面的一个特定的最小图的结的正/负交叉的数量。
We give an obstruction for genus one knots $K$, $K'$ to have the Gordian distance one by using the $0$th coefficient of the HOMFLT polynomials. As an application, we give a new constraint for genus one knot to admit a (generalized) cosmetic crossing. Combining known results, we prove the (generalized) cosmetic crossing conjecture for genus one pretzel knots.
We consider diagrams of links in $S^2$ obtained by projection from $S^3$ with the Hopf map and the minimal crossing number for such diagrams. Knots admitting diagrams with at most one crossing are classified. Some properties of these knots are exhibited. In particular, we establish which of these knots are algebraic and, for such knots, give an answer to a problem posed by Fiedler.
Crossing numbers of cable knots
利用着色琼斯多项式的次数证明恰当结的 (p,q)-缆结交叉数大于 q²c，并确定了恰当结 2-缆结及其连通和的交叉数。
Crossing number and diameter of boundary slope set of Montesinos knot
证明了边界斜率集的直径由Montesinos结的最小交叉数的两倍从上而下有界。
Knotoids are open ended knot diagrams regarded up to Reidemeister moves and isotopies. The notion is introduced by V.~Turaev in 2012. Two most important numeric characteristics of a knotoid are the crossing number and the height. The latter is the least number of intersections between a diagram and an arc connecting its endpoints, where the minimum is taken over all representative diagrams and all such an arcs disjoint from crossings. In the paper we answer the question: are there any relations between the crossing number and the height of a knotoid. We prove that the crossing number of a knot…
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。
We study relations between unknotting number and crossing number of a spatial embedding of a handcuff-graph and a theta curve. It is well known that for any non-trivial knot $K$ twice the unknotting number of $K$ is less than or equal to the crossing number of $K$ minus one. We show that this is extended to handlebody-knots. We also characterize the handlebody-knots which satisfy the equality.
Knot mosaics are used to model physical quantum states. The mosaic number of a knot is the smallest integer $m$ such that the knot can be represented as a knot $m$-mosaic. In this paper we establish an upper bound for the crossing number of a knot in terms of the mosaic number. Given an $m$-mosaic and any knot $K$ that is represented on the mosaic, its crossing number $c$ is bounded above by $(m - 2)^{2} - 2$ if $m$ is odd, and $(m - 2)^{2} - (m - 3)$ if $m$ is even.
Suppose that there exists an epimorphism from the knot group of a $2$-bridge knot $K$ onto that of another knot $K'$. In this paper, we study the relationship between their crossing numbers $c(K)$ and $c(K')$. Especially it is shown that $c(K)$ is greater than or equal to $3 c(K')$ and we estimate how many knot groups a $2$-bridge knot group maps onto. Moreover, we formulate the generating function which determines the number of $2$-bridge knot groups admitting epimorphisms onto the knot group of a given $2$-bridge knot.
We answer a question posed by Fielder in [1] concerning two notions of crossing number for algebraic knots $K$ under Hopf fibration, one topological, denoted $h(K)$, the other coming from the realization of such knots around complex singularities, denoted $C_{alg}(K)$. We show that $C_{alg}(K)-h(K)$ can be arbitrarily large. We also give an upper bound for $h$ of some families of knots such as torus knots $T(2,n)$, twist knots and their mirror images.
On the bridge number of knot diagrams with minimal crossings
给定一个节点$K$的图$D$，我们考虑交叉口的数量$c(D)$和$D$的立交桥的数量$b(D)$。我们表明，如果$D$是一个图的非平凡结$K$的交叉数$c(D)$是最小的，那么$1+\sqrt{1+c(D)} \leq b(D)\leq c(D)$。这些不等式的形式是，$b(D)$的上界是通过交替结实现的，而$b(D)$的下界是通过环面结实现的。第二个不等式只有在纽结是交错纽结时才成为等式。我们证明了第一个不等式只有当纽结是环面纽结时才成为等式。
We introduce the warping crossing polynomial of an oriented knot diagram by using the warping degrees of crossing points of the diagram. Given a closed transversely intersected plane curve, we consider oriented knot diagrams obtained from the plane curve as states to take the sum of the warping crossing polynomials for all the states for the plane curve. As an application, we show that every closed transversely intersected plane curve with even crossing points has two independent canonical orientations and every based closed transversely intersected plane curve with odd crossing points has two…
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 $K'$.
Pseudo links have two crossing types: classical crossings and indeterminate crossings. They were first introduced by Ryo Hanaki as a possible tool for analyzing images produced by electron microscopy of DNA. A normalized bracket polynomial is defined for pseudo links and then used to construct and obstruction to cosmetic crossings in classical links.
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 …
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。
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…
For a knot or link K, let L(K) be the ropelength of K and Cr(K) be the crossing number of K. In this paper, we show that there exists a constant a>0 such that L(K) is bounded above by a Cr(K) ln^5 (Cr(K)) for any knot K. This result shows that the upper bound of the ropelength of any knot is almost linear in terms of its minimum crossing number.
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 …
We study cosmetic crossings in knots of genus one and obtain obstructions to such crossings in terms of knot invariants determined by Seifert matrices. In particular, we prove that for genus one knots the Alexander polynomial and the homology of the double cover branching over the knot provide obstructions to cosmetic crossings. As an application we prove the nugatory crossing conjecture for twisted Whitehead doubles of non-cable knots. We also verify the conjecture for several families of pretzel knots and all genus one knots with up to 12 crossings.
We define and compare several natural ways to compute the bridge number of a knot diagram. We study bridge numbers of crossing number minimizing diagrams, as well as the behavior of diagrammatic bridge numbers under the connected sum operation. For each notion of diagrammatic bridge number considered, we find crossing number minimizing knot diagrams which fail to minimize bridge number. Furthermore, we construct a family of minimal crossing diagrams for which the difference between diagrammatic bridge number and the actual bridge number of the knot grows to infinity.
In this work we establish the tightest lower bound up-to-date for the minimal crossing number of a satellite knot based on the minimal crossing number of the companion used to build the satellite. If $M$ is the wrapping number of the pattern knot, we essentially show that $c(Sat(P,C))>\frac{M^2}{2}c(C)$. The existence of this bound will be proven when the companion knot is adequate, and it will be further tuned in the case of the companion being alternating.
A multi-crossing (or n-crossing) is a singular point in a projection at which n strands cross so that each strand bisects the crossing. We generalize the classic result of Kauffman, Murasugi, and Thistlethwaite, which gives the upper bound on the span of the bracket polynomial of K as 4c_2(K), to the n-crossing number: span<K> is bounded above by ([n^2/2] + 4n-8) c_n(K) for all integers n at least 3. We also explore n-crossing additivity under composition, and find that for n at least 4, there are examples of knots such that the n-crossing number is sub-additive. Further, we present the first …
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
Traditionally, knot theorists have considered projections of knots where there are two strands meeting at every crossing. A triple crossing is a crossing where three strands meet at a single point, such that each strand bisects the crossing. In this paper we find a relationship between the triple crossing number and the double crossing braid index, namely $β_2(L) \le c_3(L) + 1$. We find an infinite family of knots that achieve equality, which allows us to determine both the double crossing braid index and the triple crossing number of these knots.
Knots of Ten or Fewer Crossings of Algebraic Order Two
和谐秩序的许多代数秩序二结十个或更少的交叉点已迄今未知。我们使用卡森-戈登不变量和扭曲的亚历山大多项式，发现，在所有情况下，但一种情况下，这些结不和谐秩序二。我们还发现，一类代数阶为2的非纽结的双扭数族具有无穷协调阶。
Introduced recently, an n-crossing is a singular point in a projection of a link at which n strands cross such that each strand travels straight through the crossing. We introduce the notion of an übercrossing projection, a knot projection with a single n-crossing. Such a projection is necessarily composed of a collection of loops emanating from the crossing. We prove the surprising fact that all knots have a special type of übercrossing projection, which we call a petal projection, in which no loops contain any others. The rigidity of this form allows all the information about the knot to be …
The Crossing Number and Arc Index of Satellite Knots
本文通过弧表示和网格图方法，证明了卫星结交叉数的下界，并给出其他相关界，部分解决了K3问题列表中的问题1.2。
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