Knots and colours
本页尚未翻译成简体中文，因此以English显示。
帮助翻译 →问题
Tie a knot in a piece of string and glue the two ends together. How could you prove that it really is knotted — that no amount of wiggling, without cutting, will turn it into a plain loop?
先动手做实验，再往下读。探索没有错误的方式。
实验
用三种颜色给纽结染色
给图中的每条弧染色。在每个交叉点，相遇的三条弧必须全部同色，或者三种颜色各不相同。
点击一条弧来更换颜色（或选中它后按 Enter，或按 1、2、3）。一条弧从一个下穿处延伸到下一个下穿处。
- 此图中的交叉点
- 3
- 交叉数（可能的最少值）
- 3
已染色的弧：0 / 3。
为什么是这条规则？同一个纽结的任意两张图，都可以通过三类局部移动——赖德迈斯特移动——相互转化：加上或去掉一个扭，把一股绳滑到另一股上面，以及把一股绳滑过一个交叉点。每一种移动都把一种用了两种或更多颜色的有效染色变成另一种。
三叶结可以三染色，而平凡结不行。因为可三染色性是不变量，无论怎样摆弄，三叶结都变不成平凡结：三叶结确实是打了结的。
所以可三染色性是纽结不变量：如果一个纽结的某张图能三染色，那么它的每张图都能。一个普通的圆只有一条弧，所以平凡结不能三染色。
提示（如果你需要）
这是怎么回事
这是怎么回事
A knot diagram is a network drawing with crossings, like the ones in the dots-and-lines lesson, except that at each crossing we record which strand goes over. Two diagrams show the same knot exactly when one can be changed into the other by moving the string around — and in 1927 Kurt Reidemeister proved that it is enough to use three simple local moves: adding or removing a twist, sliding one strand over another, and sliding a strand past a crossing.
The colouring rule in the hints is called tricolouring. Each of Reidemeister's moves can be checked to keep a good colouring good (and to keep "at least two colours" true). So whether a knot can be tricoloured does not depend on how it is drawn: it is a knot invariant.
A plain loop drawn with no crossings has a single arc, so it can only ever use one colour: it is not tricolourable. The trefoil is. Therefore the trefoil is genuinely knotted — no wiggling will undo it. That is a proof, not just evidence.
Invariants have limits. The figure-eight knot cannot be tricoloured either, yet it is knotted too; showing that needs stronger invariants, such as colourings with five colours or the Jones polynomial (Vaughan Jones, 1984). Remarkably, there are algorithms that decide whether any given diagram is the unknot (Wolfgang Haken, 1961) — but whether the Jones polynomial alone can always recognise the unknot is still an open question.
由此通向何处
- 本科高年级 · 课程已上线Knot theoryKnots and links in space and their diagrams, Reidemeister moves, invariants that tell knots apart (tricolourability and Fox colourings, linking number, the Kauffman bracket and Jones polynomial), Seifert surfaces and genus, and an honest map of what is known and open.
- 研究路径 · 课程已上线Low-dimensional topology: 3- and 4-manifoldsOptional research direction: 3-manifolds through Heegaard splittings, surgery and the eight Thurston geometries; the Poincaré conjecture and geometrisation (Perelman); hyperbolic 3-manifolds; and the strange world of 4-manifolds (Freedman, Donaldson, exotic ℝ⁴), separating established results from open questions.
- 研究路径 · 课程已上线Categorical representation theoryOptional research direction: representations on categories rather than vector spaces, including tensor and fusion categories, module categories, categorical actions and categorification, such as Chuang–Rouquier sl₂-categorification and Khovanov homology.