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Knots of the Month

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December 2015

December 2015 - The Dolly Knot

This month we have chosen the Dolly knot. When undertaking LGV/HGV driver training (which one member of the SPOCK group is doing currently), you may be asked to learn, and then demonstrate in an exam, how to tie a Dolly knot. This comes as part of the training covering how to secure a load on the back of your vehicle. It is a fairly simple knot which is used in a similar way to a pulley. It allows tension to be applied to the rope easily, which in turn secures your cargo firmly. You can even use multiple numbers of Dolly knots in one piece of rope for extra effect. Furthermore, when you're finished with your Dolly knot, it is remarkably easy to remove with a bit of a tug.

There are many other, very similar, pulley-like knots used for tightening rope, all of which are classed as hitches (i.e. a knot used to bind rope to something else). In the Ashley Book of Knots, this type of knot features in knots 1514, 2124, 2125 and 2126.

November 2015

November 2015 - Knots within Proteins

This month, we have chosen to focus on knots within proteins. You might think that with proteins being incredibly long chains all curled and coiled up, they would be very prone to knotting (like earphone cables in your bag). However, it turns out that very few proteins contain knots, and when they do, they can be quite hard to spot as they are often buried deep within the complicated protein structure. When proteins do contain knots, they are often simple trefoils where one end of the chain has extended through a loop. However, a few slightly more complex knots have been found, for example see below.

Of course, one major problem with finding "knots" within proteins is that they are generally open chain structures which makes it non-trivial to define them. One common method for creating closure is to simply join the protein chain termini in a wide arc. This is easy to do if the termini are near the protein surface but difficult if they aren't as how do you decide where to draw your arc? This could end up altering how tangled your protein is. Another method is to perform a computational method which is the equivalent of grabbing the ends of the protein "string" and pulling it tight. This reduces the size of any knot and makes it easier to spot and identify. However, where you start the contraction algorithm can affect your results. An alternative is to carry out a probabilistic method involving closing the protein multiple times by ex-tending the termini to random positions on an imaginary sphere around the protein. You then collate and compare the various configurations to produce a spectrum of knots thereby identifying any knots formed as artefacts.

Whichever method you choose, there are hundreds of thousands of proteins waiting for you, so good luck! In fact, there is now even a database of proteins with knots and slipknots, called KnotProt, which even allows users to analyse their own structures.

A Group I Haloacid Dehalogenase
A Group I Haloacid Dehalogenase from Pseudomonas putida Strain PP3 (3bjxA) Contains one 61 knot as well as 31 and 41 slipknots

September 2015

September 2015 - The Shoelace Knot

This month, we chose the humble shoelace knot!

For most of us, tying our shoelaces is often done swiftly and without thought thanks to muscle memory and years of practice. But there are lots of variations on how to best tie your shoelaces, with many websites dedicated to finding the best knot for each occasion. Generally, shoelaces are tied starting with an overhand knot, or half-hitch, (which would form a trefoil if the ends were joined) and then finished off with a form of reef knot. This can be made more secure by looping around the knot twice instead of once in the final part forming a double slip knot.

If you fancy having a go at experimenting with lace tying, have a look at a site like Ian's Shoelace Site and see which becomes your favourite.

You should also see whether the overhand knot you start with is left over right, or right over left. Try each and notice how the way the final bow lays changes depending on how you started. Have you been tying your laces wrong all these years? (Of course, you could get the same effect from tying the second part of the knot in the opposite direction as well. This is because you want both stages tied in opposite directions for a smart shoelace bow, rather than in the same direction which gives a crooked bow.)

June 2015

June 2015 - Neck Tie Knot

This month, we have decided to explore neck tie knots.

One of the oldest and most commonly used knots is the four-in-hand knot which is a simple, slightly asymmetric fastening of a neck tie. Other very common tie knots are the Windsor and half-Windsor, both are which produce neat, triangular knots. Nowadays, a simple internet search reveals a multitude of websites showing increasingly fancy tie knots and their methods of creation.

In 2000, Fink and Mao, based at the Cavendish Laboratories in Cambridge, published a paper (T. M. A. Fink and Y. Mao, Physica A, 2000, 276, 109-121) in which they attempted to predict the next big thing in neck tie knots using a more rigorous approach than normal fashion trends. Around the same time the authors also wrote a commercially available book entitled "The 85 Ways to Tie a Tie" based on the work in the aforementioned paper and that of a short correspondence in Nature (T. M. A. Fink and Y. Mao, Nature, 1999, 398, 31-32). Their research is based on relating tie knots to random walks across a triangular lattice. They determine the 85 tie knots that they present in their book but say that only 13 of these are really aesthetic knots due to reasons of symmetry and balance.

This year another related paper was published, (D. Hirsch et al., PeerJ Comput. Sci., 2015, DOI: 10.7717/peerj-cs.2) this time by different authors who decided to continue the enumeration of tie knots while including knots with a textured front tied with the narrow end of the tie (something ruled out by Fink and Mao). These authors found 266,682 distinct neck tie knots which they claim are tie-able with a normal neck tie. The paper suggests that the cause of the recent proliferation of fancy tie knots was prompted by the exotic neck tie knots worn by the character of "The Merovingian" from the 2003 film The Matrix Reloaded.

May 2015

May 2015 - "Natural" Knots

It is not just humans who tie knots, knots are also found in nature and tied by animals. Here we can clearly see a Trefoil Knot and in the image on the right where we see a Granny Knot (a composition of two Trefoils of the same handedness). The likelihood is that both these trees were knotted when young by humans and have now grown into these shapes, but nethertheless we can start to think about the natural entwining of the branches and roots of trees and the complex knots these may produce.

The Ploceidae or Weaver Bird ties knots to make their nests out of a variety of materials including twigs, leaves and grass creating intricate nests to attract mates. In the video here we can see the Weaver Bird using its claws and beak to manipulate the grass, but we also see it sometimes getting the knot wrong and tying its own feet together!

Snakes are seen to tie knots in themselves, with some nice examples here of Ball Pythons curling up as small as possible. However, knots are not always good news for snakes as demonstrated in this article from National Geographi.This article discusses an "Ebola-like virus" which, when it infects a snakes, causes it to tie itself in knots and waste away!

Some animals use kntos to their advantage, such as the Hagfish. The Hagfish produces slime to escape predators and, as shown here, can tie itself in a knot (here a Trefoil) when caught by the tail to aid its escape. In this video the knot tied by the Hagfish travels from its head to its tail to escape, but Hagfish in captivity have been observed to tie a Trefoil which travels from tail to head in order to help them feed on prey. These examples show us knots are a natural thing, knot solely used by humans, and have a variety of uses.

Knotted Tree Image

April 2015

April 2015 - 3D Printing - The "Endless" (or "Eternal") Knot

This month we have been experimenting with 3D-printing knots using one of Durham University's existing printers. This is a 3D-printed version of last month's Endless Knot. Producing a physical representation of the computer-simulated knot, we gain a greater visualisation of the crossings, and a more solid knot than one made out of string, as we cannot pull the ends to tighten or deform the knot.

In this form we can easily see the knot's decorative qualities and also use the knot to aid teaching as a tactile example.

3D Print - Endless Knot

March 2015

March 2015 - The "Endless" (or "Eternal") Knot

The Endless Knot is used in many Indian and Chinese cultures in artwork, as decoration, or in ornaments. The symbol is believed to have developed from an ancient naga symbol, which depicts two intertwined snakes. It is one of the "Eight Auspicious Symbols" used by a number of Indian religions.

The "endless" nature of the Endless Knot is often seen in Buddhism (especially Tibetan Buddhism) as it symbolises many things, including the eternal continuum of the mind and the endless compassion and wisdom of the Buddha. Other interpretations include the knot being a symbol of the union of wisdom and compassion, or a visual representation of the intertwining of one's path, time and the internal Buddha. This can be linked to Buddha's teaching that, throughout the many cycles of samsara (birth, death, rebirth), at some point everyone has been your mother. Another key feature is the idea of duality, like yin and yang - the connection between two opposites, something which is mentioned in the Buddhist Dhammapada (one of the best known Buddhist scriptures).

The knot is usually drawn with 9 crossings, as shown in the picture, but if we remove the two twists at the end, we can see we get an alternating knot with 7 crossings, known as the prime knot 7_4. This knot is hyperbolic but is not amphichiral, as it differs from its mirror image. The knot can be drawn in either handedness.

Endless Knot

February 2015

February 2015 - The True Lover's Knot - A romantic theme for February

Many knots are refered to as the "True Lover's Knot", this is just one of them. The knots with this title are generally made up of two connected overhand knots, or trefoils and symbolise the connection between a couple in love. In this case the knots have been tied in two different pieces of rope, forming a link. In the Ashley Book of Knots, the "True Lover's Knot" is described in this form as being a common shape for sailor's wedding rings. The rings were made from gold and the two linked pieces could move independently but never be separated. In this case, when the ends are joined, we get a link of 12 crossings. This link is prime and has notation L12n1346, meaning it is the 1346th non-alternating link with 12 crossings. We might expect this link to be Torus, as the Trefoil Knot is, however, it is not, as it is hyperbolic.

Lover's Knot

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