内容来源：https://www.quantamagazine.org/a-simple-way-to-measure-knots-has-come-unraveled-20250922/

内容总结：

【科学前沿】困扰数学界近一个世纪的“纽结理论”猜想被推翻

在数学中，一个纽结是一段两端粘连的缠绕绳圈。如何精确衡量纽结的复杂程度，即“解结数”，是自19世纪以来困扰数学家的核心难题。1876年，苏格兰数学家彼得·格思里·泰特提出通过计算将纽结恢复为简单圆环所需的最少交叉变换次数来定义“解结数”，并希望借此区分不同纽结。然而，这一指标虽直观，实际计算却异常困难。

20世纪30年代，德国数学家希尔马尔·文特提出“可加性猜想”：两个纽结组合后的解结数应等于二者解结数之和。该猜想若成立，将意味着数学家能通过基本纽结的解结数推算出所有复合纽结的解结数，从而为纽结复杂度建立清晰秩序。近一个世纪以来，这一猜想既未被证实，也缺乏明确反例。

转折点出现在2024年。美国内布拉斯加大学数学家马克·布里滕纳姆和苏珊·赫米勒通过长达十余年的计算机辅助研究，发现一对特殊的“（2,7）环面纽结”与其镜像纽结（解结数均为3）组合后，仅需5次交叉变换即可解结，而非猜想预言的6次。这一反例不仅推翻了可加性猜想，还衍生出无限多组反例。

研究团队采用“人工搬运网络”的土法上马方式，动用数十台计算机（包括超算中心和拍卖所得的旧笔记本电脑），对数十万种纽结进行海量计算，最终从“干草堆中找出了针尖”。该成果于2024年6月发布后引发学界震动，弗吉尼亚联邦大学专家艾莉森·摩尔评价称“这一发现揭示了纽结理论中更深层的不可预测性”。

尽管结果打破了数学家对纽结世界存在有序结构的期待，但正如印第安纳大学学者查尔斯·利文斯顿所言：“它让我们意识到纽结理论比几个月前所知的更为复杂未知。”这一突破为探索纽结本质打开了新的大门，未来研究将聚焦于为何特定纽结会打破可加性规律，从而深化对复杂性的理解。

（注：本研究部分由西蒙斯基金会资助，该基金会亦资助本刊，但不影响报道独立性。）

中文翻译：

一种简单的纽结测量方法失效了

引言
1876年，彼得·格思里·泰特开始测量他称为"纽结度"的指标。这位苏格兰数学家的研究为现代纽结理论奠定了基础，他试图找到区分不同纽结的方法——这是一项众所周知的难题。在数学中，纽结是一段缠绕的绳子，两端粘合在一起。如果可以通过扭曲和拉伸（不剪断绳子）将一个纽结变成另一个，则这两个纽结相同。但仅凭外观很难判断是否可行。例如，一个看似非常复杂缠结的纽结，实际上可能等价于一个简单的环。

泰特想到了判断两个纽结是否不同的方法。首先将纽结平放在桌面上，找到绳子自身交叉的位置。剪断绳子，交换两股的位置，再重新粘合。这称为"交叉变换"。重复此操作足够多次后，最终会得到一个无结的圆环。泰特的"纽结度"就是这一过程所需的最小交叉变换次数。如今这被称为纽结的"解结数"。

若两个纽结的解结数不同，则它们必然不同。但泰特发现他的解结数带来的问题比解决的还多。"我完全陷入了思维定式"，他在给科学家朋友詹姆斯·克拉克·麦克斯韦的信中写道，"恐怕我可能遗漏或过度推崇了某些在他人看来极其简单的东西。"

如果泰特有所遗漏，那么追随他的数学家们也未能幸免。过去150年间，许多纽结理论家都被解结数所困扰。他们知道这能提供对纽结的有力描述。"这可以说是最基础的度量指标，"内布拉斯加大学的苏珊·赫米勒表示。但计算纽结的解结数往往极其困难，且该数字与纽结复杂度的对应关系常不明确。

为解开这一谜团，20世纪初的数学家提出了一个关于纽结组合时解结数变化的直接猜想。若能证明它，就能计算任意纽结的解结数——为数学家提供衡量纽结复杂度的简单具体方法。

近一个世纪的研究中，学者们几乎找不到支持或反对该猜想的证据。直到2024年6月，赫米勒与长期合作者马克·布里滕汉姆发表的论文发现：一对纽结组合后形成的纽结，其解开难度低于猜想预测。他们由此证伪了猜想，并利用反例找到了无限多对同样证伪猜想的纽结。

"论文发布时，我惊讶得叫出声，"弗吉尼亚联邦大学的艾莉森·摩尔说道。这一结果表明"解结数具有混沌性、不可预测性，是令人兴奋的研究方向"，她补充道，"论文就像在挥舞旗帜宣告：我们对此尚未理解。"

解结与未知之境

该猜想至少可追溯至1937年，德国数学家希尔马·温特开始研究纽结相加（即用同一根绳子系出两个纽结后再粘合两端）时的变化。温特认为合成纽结的解结数应始终等于两个原始纽结解结数之和。

这个被称为"可加性猜想"的预测合乎逻辑。假设将两个解结数分别为2和3的纽结相加，意味着存在2次交叉变换解开合成纽结左侧部分，3次交叉变换解开右侧部分。按此操作，总共通过2+3=5次交叉变换即可解开整个纽结。

但这仅能证明合成纽结的解结数不大于5。可能存在比分别解开两侧更高效的交叉变换序列，即实际解结数可能小于各部分之和。

要验证可加性猜想，数学家要么找到解结序列更短的合成纽结，要么证明不存在此类例子。但两者都无从下手。

部分难点在于：纽结的呈现方式（数学家称为"图解"）决定了其交叉位置和形式。同一纽结可有多种图解，要找到最短交叉变换序列，可能需要选择特定图解——通常并非常规关联的图示。

"在决定进行交叉变换前，有难以想象的大量方式可尝试调整图解，"布里滕汉姆指出，"我们最初完全无法控制图形的复杂程度。"

1985年，数学家马丁·沙勒曼取得突破，证明对于解结数为1的任意两个纽结，其合成纽结的解结数恒为2。"这使整个猜想看起来更可能成立，"印第安纳大学的查尔斯·利文斯顿表示。

该结果暗示纽结宇宙可能存在有序结构。因为所有纽结都可由更小的"素纽结"构成，可加性猜想意味着一旦掌握素纽结的解结数，就能推知所有纽结的解结数。关于任意纽结的信息都能从更简单的集合中自然推导。

"数学家希望猜想成立，因为这仿佛意味着世界存在秩序，"墨尔本大学的阿鲁尼玛·雷评论。沙勒曼的结果后来推广到其他纽结类别，但能否适用于所有纽结仍不明确。

此时布里滕汉姆和赫米勒调动了计算机集群助力。

人工搬运网络

十年前二人启动该项目时，目标更为宏大：利用计算机探索解结数的各种特性。他们采用名为SnapPy的软件，该软件运用复杂几何技术判断两幅图示是否代表同一纽结。此前SnapPy数据库大幅扩容，已能识别近6万种独特纽结。

这完美契合他们的需求。他们从单个复杂纽结出发，施加所有可能的交叉变换生成大量新纽结，再用SnapPy识别这些纽结——如此循环往复。

他们对数百万个对应数十万种纽结的图解进行此操作，最终构建了关于解结序列的庞大信息库，计算出数千种纽结解结数的上限。研究需要巨大算力：他们申请了内布拉斯加大学计算中心的超算资源，同时在拍卖购得的旧笔记本电脑上运行程序，共管理数十台设备。"我们建立了人工搬运网络，"布里滕汉姆解释道，"靠人在计算机间走动传输数据。"

这套系统持续运行逾十年。期间部分设备因过热甚至起火损毁。"有台电脑真的冒出了火花，"布里滕汉姆回忆，"还挺有趣的。"（他补充这些机器已"光荣退役"。）

2024年秋，一篇关于机器学习证伪可加性猜想失败的论文引起二人注意。他们意识到机器学习可能非最佳方案："反例如同大海捞针，"赫米勒说，"这并非机器学习所长，后者更擅长寻找规律。"但这强化了他们的预感：精心打造的人工搬运网络或许能找到针尖。

纽带相连

二人意识到可利用已发现的解结序列寻找可加性猜想的潜在反例。假设有两个解结数分别为2和3的纽结，试图解开其合成纽结。一次交叉变换后得到新纽结。若猜想成立，原纽结解结数应为5，新纽结应为4。

但如果已知新纽结解结数实际为3呢？这意味着原纽结仅需四步即可解开，从而打破猜想。

"这些中间纽结能告诉我们什么？"布里滕汉姆说。他们已有完美工具：耗时十年构建的数据库，包含数千种纽结解结数上限。数学家开始配对叠加纽结，逐次分析其合成纽结的解结序列。他们重点关注那些解结数估值范围宽泛的合成纽结，但待排查列表仍极其庞大——"绝对数千万，可能上亿，"布里滕汉姆表示。

数月间，计算机程序对这些纽结进行交叉变换，并将结果与数据库比对。暮春某日，布里滕汉姆照常检查输出文件时，意外发现一行文字："合成纽结猜想不成立"。这是他们预设的代码提示——但从未期待真正出现。

最初他们怀疑结果。"第一反应是程序出错，"布里滕汉姆说。"我们放下一切事务，"赫米勒回忆，"生活完全停滞，连吃饭睡觉都变得烦人。"但程序验证无误。他们甚至用绳子打出该纽结，手动完成解结流程以确认。反例真实存在。

缠绕之谜

他们发现的反例由两个(2,7)环面纽结复制体构成。这种纽结由两股绳子缠绕三周半后对接两端形成，其镜像版本反向缠绕三周半构成。原纽结与其镜像的解结数均为3。但程序显示：将这两类纽结相加后，仅需五步即可解开结果——而非猜想预测的六步。

"这是个惊人简单的反例，"摩尔评价，"回归了交叉变换的不可预测性。"该结果引导二人发现无限多反例，涵盖几乎所有由两股缠绕粘合构成的纽结。

随着可加性猜想被推翻，纽结理论界迎来广阔探索空间。对某些数学家而言，这一结果令人失望，揭示出纽结世界的结构比预期更少。"解结数不如我们期望的规整，"雷表示，"这有点遗憾。"但另一方面，这反而增加了解结数的魅力。"纽结理论的复杂性和未知领域比几个月前所知更为深广，"利文斯顿说。

这种复杂性的本质尚不明确。在紧张验证反例期间，布里滕汉姆和赫米勒未能形成关于该反例为何能打破猜想的直觉。理解这一点将帮助数学家更好地把握纽结复杂度的成因。

"我仍被这个关于解结数的最基本问题难住，"摩尔说，"而这正是激励我们前进的动力。"

编者注：布里滕汉姆与赫米勒的研究部分由西蒙斯基金会资助，本独立编辑杂志亦获该基金会支持。西蒙斯基金会的资助决策不影响本刊报道内容。

英文来源：

A Simple Way To Measure Knots Has Come Unraveled
Introduction
In 1876, Peter Guthrie Tait set out to measure what he called the “beknottedness” of knots.
The Scottish mathematician, whose research laid the foundation for modern knot theory, was trying to find a way to tell knots apart — a notoriously difficult task. In math, a knot is a tangled piece of string with its ends glued together. Two knots are the same if you can twist and stretch one into the other without cutting the string. But it’s hard to tell if this is possible based solely on what the knots look like. A knot that seems really complicated and tangled, for instance, might actually be equivalent to a simple loop.
Tait had an idea for how to determine if two knots are different. First, lay a knot flat on a table and find a spot where the string crosses over itself. Cut the string, swap the positions of the strands, and glue everything back together. This is called a crossing change. If you do this enough times, you’ll be left with an unknotted circle. Tait’s beknottedness is the minimum number of crossing changes that this process requires. Today, it’s known as a knot’s “unknotting number.”
If two knots have different unknotting numbers, then they must be different. But Tait found that his unknotting numbers generated more questions than they answered.
“I have got so thoroughly on one groove,” he wrote in a letter to a friend, the scientist James Clerk Maxwell, “that I fear I may be missing or unduly exalting something which will appear excessively simple to anyone but myself.”
If Tait missed something, so did every mathematician who followed him. Over the past 150 years, many knot theorists have been baffled by the unknotting number. They know it can provide a powerful description of a knot. “It’s the most fundamental [measure] of all, arguably,” said Susan Hermiller of the University of Nebraska. But it’s often impossibly hard to compute a knot’s unknotting number, and it’s not always clear how that number corresponds to the knot’s complexity.
To untangle this mystery, mathematicians in the early 20th century devised a straightforward conjecture about how the unknotting number changes when you combine knots. If they could prove it, they would have a way to compute the unknotting number for any knot — giving mathematicians a simple, concrete way to measure knot complexity.
Researchers searched for nearly a century, finding little evidence either for or against the conjecture.
Then, in a paper posted in June, Hermiller and her longtime collaborator Mark Brittenham uncovered a pair of knots that, when combined, form a knot that is easier to untie than the conjecture predicts. In doing so, they disproved the conjecture — and used their counterexample to find infinitely many other pairs of knots that also disprove it.
“When the paper was posted, I gasped out loud,” said Allison Moore of Virginia Commonwealth University.
The result demonstrates that “the unknotting number is chaotic and unpredictable and really exciting to study,” she added. The paper is “like waving a flag that says, we don’t understand this.”
Unknotting and the Great Unknown
The conjecture dates back to at least 1937, when the German mathematician Hilmar Wendt set out to understand what happens when you add knots together — that is, when you tie both of them with the same string before gluing the ends together. (Mathematicians call this combined knot the “connect sum.”) Wendt thought that the unknotting number of the resulting knot should always be the sum of the unknotting numbers of the two original knots.
His prediction, now known as the additivity conjecture, makes sense. Say you add the two knots above, whose unknotting numbers are known to be 2 and 3. That means that there’s a sequence of two crossing changes that unknots the lefthand side of the connect sum, and a sequence of three crossing changes that unknots the righthand side. If you use these sequences, you can unknot the whole thing in 2 + 3, or 5, crossing changes.
But this only tells you that the connect sum’s unknotting number is no bigger than 5. You might be able to find a sequence of crossing changes that’s more efficient than untying each side individually. That is, there might be a knot that really is less than the sum of its parts.
To settle the additivity conjecture, mathematicians had to either find a connect sum with a shorter unknotting sequence or prove that no such example exists. In either case, they didn’t have a clue where to begin.
Part of the problem was that the way you lay out your knot — what mathematicians call a “diagram” — determines where and how the knot crosses over itself. There are lots of diagrams that can represent the same knot. To find the shortest sequence of crossing changes, you might have to choose just the right diagram. Often, it’s not the one you’d normally associate with the knot.
“There are unimaginably large numbers of ways to try and imagine changing your diagram before you decide to introduce the crossing change,” Brittenham said. “We don’t, at least at the start, have any control over how complicated the picture has to look.”
In 1985, the mathematician Martin Scharlemann finally made some headway when he proved that for any two knots whose unknotting number is 1, the connect sum will always have an unknotting number of 2. “That made [the whole conjecture] seem much more likely,” said Charles Livingston of Indiana University.
The result offered tantalizing evidence that the universe of knots could be neatly organized. That’s because all knots can be built out of a smaller class of “prime” knots. The additivity conjecture implied that once you knew the unknotting numbers of those prime knots, you would know them for all knots. Any information you might want about a given knot would fall naturally out of that much simpler set.
Mathematicians wanted the conjecture to be true, said Arunima Ray of the University of Melbourne, “because that would be like, there’s order in the world.”
Scharlemann’s result was later extended to other classes of knots. But it wasn’t clear that it would apply to all knots.
Then Brittenham and Hermiller convened a cluster of computers to help.
Sneakernet
The pair began their project a decade ago with a broader aim: to use computers to learn whatever they could about the unknotting number.
They turned to software known as SnapPy, which uses sophisticated geometric techniques to test whether two pictures depict the same knot. Just a few years earlier, SnapPy had vastly expanded its database, enabling it to identify nearly 60,000 unique knots.
It was perfectly suited for what Brittenham and Hermiller had in mind. They started with a single complicated knot and applied every imaginable crossing change to it, producing scores of new knots. They then used SnapPy to identify those knots — and repeated the process.
They did this for millions of knot diagrams that corresponded to hundreds of thousands of knots. Ultimately, they assembled an enormous library of information about unknotting sequences and calculated upper bounds on the unknotting numbers of thousands of knots. The work required a lot of computing power: The pair signed up for supercomputing time at the University of Nebraska’s computing center, while also running their program on old laptops they’d bought at an auction. All told, they were managing dozens of computers. “We had a bit of a sneakernet,” Brittenham said, “where you transfer information from computer to computer by walking between them.”
The duo kept their program running in the background for over a decade. During that time, a couple of computers from their ragtag collection succumbed to overheating and even flames. “There was one that actually sent out sparks,” Brittenham said. “That was kind of fun.” (Those machines, he added, were “honorably retired.”)
Then, in the fall of 2024, a paper about a failed attempt to use machine learning to disprove the additivity conjecture caught Brittenham and Hermiller’s attention. Perhaps, they thought, machine learning wasn’t the best approach for this particular problem: If a counterexample to the additivity conjecture was out there, it would be “a needle in a haystack,” Hermiller said. “That’s not quite what things like machine learning are about. They’re about trying to find patterns in things.”
But it reinforced a suspicion the pair already had — that maybe their more carefully honed sneakernet could find the needle.
The Tie That Binds
Brittenham and Hermiller realized they could make use of the unknotting sequences they’d uncovered to look for potential counterexamples to the additivity conjecture.
Imagine again that you have two knots whose unknotting numbers are 2 and 3, and you’re trying to unknot their connect sum. After one crossing change, you get a new knot. If the additivity conjecture is to be believed, then the original knot’s unknotting number should be 5, and this new knot’s should be 4.
But what if this new knot’s unknotting number is already known to be 3? That implies that the original knot can be untied in just four steps, breaking the conjecture.
“We get these middle knots,” Brittenham said. “What can we learn from them?”
He and Hermiller already had the perfect tool for the occasion humming away on their suite of laptops: the database they’d spent the previous decade developing, with its upper bounds on the unknotting numbers of thousands of knots.
The mathematicians started to add pairs of knots and work through the unknotting sequences of their connect sums. They focused on connect sums whose unknotting numbers had only been approximated in the loosest sense, with a big gap between their highest and lowest possible values. But that still left them with a massive list of knots to work through — “definitely in the tens of millions, and probably in the hundreds of millions,” Brittenham said.
For months, their computer program applied crossing changes to these knots and compared the resulting knots to those in their database. One day in late spring, Brittenham checked the program’s output files, as he did most days, to see if anything interesting had turned up. To his great surprise, there was a line of text: “CONNECT SUM BROKEN.” It was a message he and Hermiller had coded into the program — but they’d never expected to actually see it.
Initially, they were doubtful of the result. “The very first thing that went through our heads was there was something wrong with our programming,” Brittenham said.
“We just dropped absolutely everything else,” Hermiller recalled. “All of life just went away. Eating, sleeping got annoying.”
But their program checked out. They even tied the knot it had identified in a rope, then worked through the unknotting procedure by hand, just to make sure.
Their counterexample was real.
Twisted Mysteries
The counterexample Brittenham and Hermiller found is built out of two copies of a knot called the (2, 7) torus knot. This knot is made by winding two strings around each other three and a half times and then gluing their opposing ends together. Its mirror image is made by winding three and a half times in the other direction.
The unknotting number of both the (2, 7) torus knot and its mirror image is 3. But Brittenham and Hermiller’s program found that if you add these knots, you can unknot the result in just five steps — not six, as the additivity conjecture predicted.
“It’s a shockingly simple counterexample,” Moore said. “It goes back to that unpredictability of the crossing change.”
The result led Brittenham and Hermiller to an infinite list of other counterexamples, including almost any knot that’s built by winding two strings and gluing.
Now, with the additivity conjecture decisively struck down, the knot theory community has a wide world to explore.
For some mathematicians, the new result brings disappointment. It reveals that there’s less structure in the world of knots than they had hoped for. The unknotting number is “not as well behaved as we would like,” Ray said. “That’s a bit sad.”
But from another perspective, that only makes the unknotting number more intriguing. “There’s just much more complexity and unknowns about knot theory than we knew there were a few months ago,” Livingston said.
The nature of that additional complexity isn’t clear yet. During their furious examination of their counterexample, Brittenham and Hermiller weren’t able to develop an intuition for why it broke the additivity conjecture when other knots didn’t. Understanding this could help mathematicians get a better handle on what makes some knots complex and others less so.
“I’m still stymied by this most basic question” about the unknotting number, Moore said. “That just lights the fire under you.”
Editor’s Note: Brittenham and Hermiller’s research was funded in part by the Simons Foundation, which also funds this editorially independent magazine. Simons Foundation funding decisions have no influence on our coverage.