解开数量和解开链接所需的Reidemeister移动数量

不规则平滑和Reidemeister移动次数

Irregular smoothing and the number of Reidemeister moves

在前面的文章中，我们考虑了一个Hass和Nowik型的链接图不变量，使用正则平滑和unknotting数，估计Reidemeister移动所需的解开。在本文中，我们引入了一个新的链接图不变量使用不规则平滑，并给出了一个例子的纽结图的unknot新的不变量给出了一个更好的估计比旧的。

Automated Reidemeister Moves: A Numerical Approach to the Unknotting Problem

In mathematics, a knot is a single strand of string crossed over itself any number of times, and connected at the ends. The Reidemeister Moves have been proven to be the three core moves necessary to fully untangle a knot. Some knots can be untangled to a loop (the unknot), while others are fundamentally knotted. We define four generalized moves based on the Reidemeister Moves. These moves have the capability of untangling knots which must be made more complicated before they can be simplified. With these moves, we construct a computer program which reads the two-dimensional projection of a kn…

An Index-Type Invariant of Knot Diagrams Giving Bounds for Unknotting Framed Unknots

We introduce a new knot diagram invariant called the Self-Crossing Index (SCI). Using SCI, we provide bounds for unknotting two families of framed unknots. For one of these families, unknotting using framed Reidemeister moves is significantly harder than unknotting using regular Reidemeister moves. We also investigate the relation between SCI and Arnold's curve invariant St, as well as the relation with Hass and Nowik's invariant, which generalizes cowrithe. In particular, the change of SCI under Ω3 moves depends only on the forward/backward character of the move, similar to how the change of …

The unbearable hardness of unknotting

We prove that deciding if a diagram of the unknot can be untangled using at most $k$ Riedemeister moves (where $k$ is part of the input) is NP-hard. We also prove that several natural questions regarding links in the $3$-sphere are NP-hard, including detecting whether a link contains a trivial sublink with $n$ components, computing the unlinking number of a link, and computing a variety of link invariants related to four-dimensional topology (such as the $4$-ball Euler characteristic, the slicing number, and the $4$-dimensional clasp number).

Unknotting twisted knots with Gauss diagram forbidden moves

Twisted knot theory, introduced by M.O.Bourgoin, is a generalization of virtual knot theory. It is easily shown that any virtual knot can be deformed into a trivial knot by a finite sequence of generalized Reidemeister moves and two "forbidden moves" $F_{1}$ and $F_{2}$. Similarly, we show that any twisted knot also can be deformed into a trivial knot or a trivial knot with a bar by a finite sequence of extended Reidemeister moves and three "forbidden moves" $T_{4}$, $F_{1}$ (or $F_{2}$) and $F_{3}$ (or $F_{4}$) .

H(n)-移动是虚拟和焊接链环的解结操作

The H(n)-move is an unknotting operation for virtual and welded links

本文将经典纽结的 H(n)-解结移动推广至虚拟与焊接链环，并证明虚拟化和禁止移动均可由有限次广义 Reidemeister 移动与 H(n)-移动实现。

用高斯图禁忌步解开虚结

Unknotting virtual knots with Gauss diagram forbidden moves

禁止的移动可以与高斯图Reidemeister移动相结合，以获得移动序列，我们可以将任何高斯图（因此任何虚拟结）改变为任何其他，特别是包括未结图

经典链接和焊接链接的一种新无结操作

A new unknotting operation for classical and welded links

本文提出一种新的无结操作，能够将任何结或链简化为无结或无链，通过有限的无结操作和 Reidemeister 操作，并分析无结操作下的结距离。

用Reidemeister步实现矩形图上的外克伦威尔步

Realizing Exterior Cromwell moves on rectangular diagrams by Reidemeister moves

如果一个矩形图表示平凡纽结，那么它可以通过一个有限序列的合并操作和交换操作变形为只有四条边的平凡矩形图，而不需要增加边的数目，如I. A.戴尼科夫利用这一点，亨利克和考夫曼给出了解开琐碎结的结图所需的里德迈斯特移动次数的上界。然而，交换或合并移动的顶部和底部对矩形图的边缘没有考虑在证明。本文证明了平凡纽结存在一个矩形图，它需要这样的交换移动才能解开，并研究了实现这样的交换或合并移动所需的Reidemeister移动数的上界.

纽结图与其解结数的关系

The Relation Between Diagrams of a Knot and Its Unknotting Number

解纽结数是纽结的经典不变量。然而，一般来说，它的确定是困难的。为了从定义中得到解结数，必须研究所有可能的纽结图。我们试图证明解结数可以从任何一个图的结。为此，我们试图证明解结数在Riedemiester移动下不变，但这样的命题是不正确的。Reidemeister II移动可以改变解开号码。参见Nakanishi-Bleiler示例。所以这篇文章被撤回了。

纽结理论中自然出现的NP困难问题

NP-hard problems naturally arising in knot theory

本文证明纽结理论中若干自然问题（如有限步Reidemeister移动转换、解链数判定、寻找子链环等）为NP困难或NP完全问题。

结简化强化学习器、难结与结简化数

RL unknotter, hard unknots and unknotting number

本文提出一种强化学习管道用于简化结图，通过训练代理学习移动提议和价值启发式方法，测试了非常难的无结图，并通过图膨胀方法在4_1#9_{10}上恢复了结简化数的上界为三。

Gordian距离与链接的钳子手术

Gordian distance and clasper surgery for links

本文证明，除了环绕数外，任何链接通过最多n^2次交叉变化可变为C_k平凡链接，从而表明Milnor不变量对解链数信息有限。