Conformal invariance of the writhe of a knot

The Writhe of Permutations and Random Framed Knots

We introduce and study the writhe of a permutation, a circular variant of the well-known inversion number. This simple permutation statistics has several interpretations, which lead to some interesting properties. For a permutation sampled uniformly at random, we study the asymptotics of the writhe, and obtain a non-Gaussian limit distribution. This work is motivated by the study of random knots. A model for random framed knots is described, which refines the Petaluma model. The distribution of the framing in this model is equivalent to the writhe of random permutations.

12交叉结的绳长与空间缠绕数（writhe）量子化

Ropelength and writhe quantization of 12-crossing knots

本文测量了2176个12交叉结的绳长，发现紧致交错结和非交错结的空间缠绕数分别呈现接近4/7和4/3倍数的量子化现象，并揭示了绳长与双曲体积、空间缠绕数与Rasmussen s不变量的正相关性。

Subspaces of Knot Spaces

The inclusion of the space of all knots of a prescribed writhe in a particular isotopy class into the space of all knots in that isotopy class is a weak homotopy equivalence.

相交图与拧数多项式

Intersection graph and writhe polynomial

本文建立虚拟纽结相交图等价性与拧数多项式相等之间的充要条件，揭示二者的一一对应关系。

Typical knots: size, link component count, and writhe

We model the typical behavior of knots and links using grid diagrams. Links are ubiquitous in the sciences, and their "normal" or "typical" behavior is of significant importance in understanding situations such as the topological state of DNA or the statistical mechanics of ring polymers. We examine three invariants: the expected size of a random knot; the expected number of components of a random link; and the expected writhe of a random knot. We investigate the first two numerically and produce generating functions which codify the observed patterns: knot size is uniformly distributed and li…

Real algebraic knots of low degree

In this paper we study rational real algebraic knots in $\R P^3$. We show that two real algebraic knots of degree $\leq5$ are rigidly isotopic if and only if their degrees and encomplexed writhes are equal. We also show that any irreducible smooth knot which admits a plane projection with less than or equal to four crossings has a rational parametrization of degree $\leq 6$. Furthermore an explicit construction of rational knots of a given degree with arbitrary encomplexed writhe (subject to natural restrictions) is presented.

An unknotting index for virtual knots

In this paper we introduce the notion of an unknotting index for virtual knots. We give some examples of computation by using writhe invariants, and discuss a relationship between the unknotting index and the virtual knot module. In particular, we show that for any non-negative integer $n$ there exists a virtual knot whose unknotting index is $(1,n)$.

$Ξ$-移动下$2$-分支虚链环的分类

Classification of $2$-component virtual links up to $Ξ$-moves

本文通过改进奇缠绕数和链环数，将Taniguchi等人关于$Ξ$-移动刻画虚纽结奇缠绕数的结果扩展，完成了$Ξ$-移动下$2$-分支虚链环的分类。

On arrow polynomials of checkerboard colorable virtual links

In this paper we give two new criteria of detecting the checkerboard colorability of virtual links by using odd writhe and arrow polynomial of virtual links, respectively. By applying new criteria, we prove that 6 virtual knots are not checkerboard colorable, leaving only one virtual knot whose checkerboard colorability is unknown among all virtual knots up to four classical crossings.

关于手柄体中交替结的注记

A note on alternating knots in handlebodies

本文研究手柄体中交替结的最小交叉数和旋转数，并推广Jones多项式到该设置中。

On the crossing number of positive knots and braids and braid index criteria of Jones and Morton-Williams-Franks

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…

Mutation invariance of the arc index for some Montesinos knots

For the alternating knots or links, mutations do not change the arc index. In the case of nonalternating knots, some semi-alternating knots or links have this property. We mainly focus on the problem of mutation invariance of the arc index for nonalternating knots which are not semi-alternating. In this paper, we found families of infinitely many mutant pairs/triples of Montesinos knots with the same arc index.

Möbius invariance of knot energy

A physically natural potential energy for simple closed curves in $\bold R^3$ is shown to be invariant under Möbius transformations. This leads to the rapid resolution of several open problems: round circles are precisely the absolute minima for energy; there is a minimum energy threshold below which knotting cannot occur; minimizers within prime knot types exist and are regular. Finally, the number of knot types with energy less than any constant $M$ is estimated.

The invariance of knot lattice homology

Assume $Γ$ is a negative-definite forest with exactly one unframed vertex, and $M(Γ)$ is the resulting plumbed 3-manifold with a knot embedded. We show that the filtered lattice chain homotopy type of $Γ$ is an invariant of the diffeomorphism type of $M(Γ)$.

纽结格同调与同伦的不变性和自然性

Invariance and naturality of knot lattice homology and homotopy

该研究证明纽结格同调是有理同调球中广义代数纽结光滑型的不变量，可由双重滤过同伦型的胞腔同调实现，还揭示了其同伦的自然性及奇点分解拓扑的相关结论。

学习拓扑不变性

Learning Topological Invariance

提出自动化学习拓扑不变性的机器学习方法，将其应用于纽结理论，通过对比与生成技术映射同一纽结类表示，还开发师生框架解释学习的不变量，并探索解决琼斯无结猜想的方法。

Invariance and the knot Floer cube of resolutions

This paper considers the invariance of knot Floer homology in a purely algebraic setting, without reference to Heegaard diagrams, holomorphic disks, or grid diagrams. We show that (a small modification of) Ozsváth and Szabó's cube of resolutions for knot Floer homology, which is assigned to a braid presentation with a basepoint, is invariant under braid-like Reidemeister moves II and III and under conjugation. All moves are assumed to happen away from the basepoint. We also describe the behavior of the cube of resolutions under stabilization. The techniques echo those employed to prove the inv…

A combinatorial proof of invariance of double-point enhanced grid homology

We prove that the "minus" version of Lipshitz's double-point enhanced grid homology is a knot invariant through purely combinatorial means.

On ambiguity in knot polynomials for virtual knots

We claim that HOMFLY polynomials for virtual knots, defined with the help of the matrix-model recursion relations, contain more parameters, than just the usual $q$ and $A = q^N$. These parameters preserve topological invariance and do not show up in the case of ordinary (non-virtual) knots and links. They are most conveniently observed in the hypercube formalism: then they substitute $q$-dimensions of certain fat graphs, which are not constrained by recursion and can be chosen arbitrarily. The number of these new topological invariants seems to grow fast with the number of non-virtual crossing…

On the classification of rational tangles

This paper gives two new combinatorial topological proofs of the classification of rational tangles. Each proof rests on an elegant lemma showing that rational tangles are isotopic to canonical alternating rational tangles. The first proof defines the tangle fraction from the canonical form and uses flyping to prove invariance. The second proof defines the fraction of a rational tangle via integral coloring of the tangle. The coloring method is then used to prove the Kauffman-Harary coloring conjecture for alternating knots and links in the case of rational knots and links (closures of rationa…

结三角剖分的积分面-边对偶性

Integral face-edge duality for knot triangulations

本文通过上链构造建立结三角剖分中面矩阵与粘合矩阵的对偶关系，证明零化度加倍及FAMED恒等式，并给出Wong猜想的拟议证明。

On Conway mutation and link homology

We give a new, elementary proof that Khovanov homology with $\mathbb{Z}/2\mathbb{Z}$--coefficients is invariant under Conway mutation. This proof also gives a strategy to prove Baldwin and Levine's conjecture that $δ$--graded knot Floer homology is mutation--invariant. Using the Clifford module structure on $\widetilde{\text{HFK}}$ induced by basepoint maps, we carry out this strategy for mutations on a large class of tangles. Let $L'$ be a link obtained from $L$ by mutating the tangle $T$. Suppose some rational closure of $T$ corresponding to the mutation is the unlink on any number of compon…

交错纽结与链环的CWR不变量的无平方因子矩阵公式

Squarefree Matrix Formulas for the CWR Invariant of Alternating Knots and Links

该研究针对有向非分裂交错链环的CWR不变量，提出基于合并Tait图的统一无平方因子迹构造，得到任意k≥3的CWR_k公式及相关生成多项式、反演公式，并推导出CWR4、CWR5的显式公式与奇分量消失判据等结果。