Disentangling Loops with an Ambient Isotopy


Can one of these shapes be continuously distorted to produce the other?

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 S¹ to R³. Two knots are equivalent if one can be distorted into the other without breaking it. More formally, two knots K1 and K2 are equivalent if K1 can be continuously distorted into K2 by a family of 1-to-1 continuous maps (with continuous inverses) from R³ to R³. 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 R³ 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.


Left: The Unknot. Right: The Trefoil Knot [Image Wikimedia]

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.


Dextro and Laevo versions of Trefoil Knot. {Wikimedia]


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.


Left to right: Unlink, Hopf Link, Borromean Rings.

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!


Figure from Wells, 1991.


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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