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Impossible Knots

Knot theory is an important twentieth-century branch of mathematics with close relations to topology. It aims to analyse and distinguish the infinite number of knots into classes of equivalent knots, and there are indeed an infinite number of such equivalence classes of equivalent knots.

Early on in the present study of impossible images, it seemed clear that there were significant connections with knot theory. The connections were not direct, it must be stressed, because knots are floppy things, whereas the effects we see in at least the Necker cubes and the triangles depend on the sides being interpreted by the brain as straight lines, and the faces being interpreted as planar. This means that the phenomena we are dealing with, or at least some of them, are metrical and/or projective, not as general as topological. Nonetheless, there are two striking facts about knots that relate them to impossible images.

First, note that knots are three-dimensional objects, but they are represented in two dimensions, on the page, by knot-diagrams. Also, while a real knot is a thick thing, for mathematical purposes it is assumed to have no cross-section, that is to be homeomorphic to the 1-dimensional circle S1. Its knot properties are not changed by this simplifying assumption. But there are many ways to be homeomorphic to S1, ways that are not equivalent as knots. Here, I will (generally) draw knots as they are in reality, that is made with a thick cord.

Knots are conventionally assumed to have their ends joined up, so that they cannot be changed without cutting or breaking the join. Below is an image of the knot-diagram of simplest kind of non-trivial knot, the trefoil. (To say that a knot is non-trivial is to say that it cannot be transformed by untying to the circle S1, without cutting.). Correspondingly, the trefoil is the simplest knot to tie, given a single piece of unknotted rope.

You will notice three occlusions in this knot diagram, where one part of the knot passes over another. The occlusions are conventionally represented by having the underneath part of the knot broken and the on-top part unbroken. But this is precisely the device which suggested itself to disambiguate the Necker cubes, and to produce the inconsistent Necker cube shown on the beginning page of this website. Thus, the primary representational device in knot theory, occlusions represented by broken lines, found a ready application in inconsistent images. Moreover, since it is the representation of occlusion that is relevant here, it suggests that impossible images have a lot to do with occlusions. This is what led to the conjecture that all impossible images are occlusion-illusions. This conjecture is clearly true of some cases, but interestingly it is not obvious for all, particularly the Schuster fork and the Ernst stairs: some other functions of the brain must be responsible for those.

The second connection with knot theory came when we realised that there was a relationship between the trefoil and the Reutersvard/Penrose triangle, in particular the latter can be inscribed in the former. In what its discover, Leishman, calls the Ghent Knot, a sheet of paper with the three dimensional representation of a trefoil knot on one side and the Reutersvard/Penrose triangle strategically drawn on the other side presents itself as a smooth transition from one to the other by simply holding the sheet of paper up to the light and back again (see movie). The salient steps in this transformation are seen in the 3 pictures below.

One can be transformed into the next by addition or subtraction. The middle picture illustrates a way of making the trefoil inconsistent. Interestingly, the inconsistent triangle has turned-in (occluding) corners which just match the occlusions of the trefoil. In fact the inside of the trefoil amounts to a 3-twist Mobius strip, which immediately gives the (informal) result that the inner surface of a Penrose triangle is a 3-twist Mobius strip.

There are also knot-like objects which are flat (in three dimensions). The simplest example is a dog collar (two edges and two sides). A corresponding trefoil-like object can be obtained by cutting, tying, then rejoining; but not without cutting. A dog collar is distinct from a Mobius strip (one edge and one side), and a distinct but corresponding trefoil-like object can be obtained by cutting, tying then rejoining; but not without cutting.

This also raises the question of whether an impossible knot can be drawn. The above (middle) image is of course an inconsistent picture which is a trefoil knot, but its inconsistency is in a sense incidental. The triangle could be substituted by a consistent triangle without essential modification. That is, the trefoil isn’t inconsistent as a knot. Even so, the middle knot is clearly a trefoil, which indicates that the classical consistent theory of knots ought to go over to the inconsistent case.

Eventually, Steve Leishman used the principle of the Schuster fork to construct what I contend is a genuine inconsistent knot.

The fact that this can be done for structures like knots only by using the Schuster method, reinforces the suspicion that the inconsistency of Schuster is topological, not metrical nor projective. Note in passing that each of the three crossings represents a different way that the occlusion can be drawn; of course it is straightforward to make these the same as one another.

There is another way in which the challenge to produce an inconsistent knot can be interpreted, namely to inconsistently tie a (consistent) knot. For example, if a piece of string is held in both hands, then a trefoil can’t be turned into a trivial knot, or vice versa, without letting go at some stage. So, to turn a trivial knot into a trefoil without letting go is to do something inconsistent. Allez:

Voila! I must stress in all honesty that I don’t let go of the string with either hand as I tie it. Of course there’s a trick, but there is also an interesting philosophical question here, of what counts as “not letting go”. Think about it.

Text: Chris Mortensen
Images: Steve Leishman, Chris Mortensen, Peter Quigley
June 2007