The surface in the left panel above has two linked loops. In the right hand panel, the loops are unlinked. Is it possible to continuously distort the left-hand surface so as to unlink the loops and produce the right-hand figure? This seems impossible, but intuition is not always reliable.

**Knot Theory**

A knot is an embedding of a circle into 3-space, that is, a homeomorphism or bicontinuous bijection from **S****¹** to **R****³**. Two knots are equivalent if one can be distorted into the other without breaking it. More formally, two knots **K _{1}** and

**K**are equivalent if

_{2}**K**can be continuously distorted into

_{1}**K**by a family of 1-to-1 continuous maps (with continuous inverses) from

_{2}**R**

**³**to

**R**

**³**. The whole space is distorted, and the knot must not intersect itself during the distortion process. This kind of mapping is called an

**ambient isotopy**, a continuous distortion of space that maps a submanifold to another submanifold. The whole ambient space

**R**

**³**is distorted and the embedded knot “hitches a ride”.

**The Unknot and Trefoil**

The simplest knot, a smooth distortion of the circle, is called the **unknot**. Another simple example is the trefoil knot. It is impossible to distort the trefoil continuously without self-intersection so as to obtain the unknot. The two knots are not equivalent.

The trefoil comes in two flavours, each a mirror image of the other. They are not equivalent: it is impossible (in 3-space) to distort one version smoothly and continuously into the other.

**Links**

A link is a set of disconnected components, each of which is a knot. The simplest example is the delightfully-named **unlink**. The simplest non-trivial link is the Hopf Link, with each of two unknots encircling the other. A more complex example is the set of Borromean Rings, with three components: no two components are linked, but the trio is inextricably intertwined.

**Back to the Original Question**

Can we continuously distort the left hand figure below to produce the right-hand one?

The surprising answer is that we can. The figure below shows a sequence of five surfaces. It is clear that each successive surface is a continuous distortion of the preceding one. But the first and last surfaces correspond to the two surfaces above, so they are **ambient isotopic** or, more simply, equivalent.

**Undoing Handcuffs**

Another surprising example is given in David Wells’ book *The Penguin Dictionary of Curious and Interesting Geometry*. It shows a “pair of hand-cuffs” looped around a circle. By the continuous sequence of distortions shown in the figure, it appears that one of the cuffs is unlinked!

**Sources**

Cromwell, Peter, 2004: *Knots and Links.* Cambridge University Press, 328pp.

Wells, David, 1991: *The Penguin Dictionary of Curious and Interesting Geometry*. Penguin Books.

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