Research


Culture Conference 2015, Birmingham - Knot Tying Exercise


Please see below the results of the knot tying exercise at Culture Conference 2015, Birmingham

Knot 8.21

The knot participants were asked to tie was a stopper knot, used generally at the end of a piece of rope to stop the rope unravelling, passing back through a hole or slipping through another knot. By joining up the ends, this knot can be characterised as knot 8_21 (seen here on KnotAtlas).

Participants were given the picture (shown left) and asked to tie the knot. They could have as much time and attempts as they wanted and did not have to submit a knot if they did not want to. Eight knots were produced as part of this experiment - more may have been attempted but given up! Of the eight knots tied, four were the correct knot, giving us a 50% success rate in this case! The other knots tied were; two cases of the knot 5_1 (see here) one case of the knot 5_2 (see here) one case of the knot 6_2 (see here)




Knot 5.1

The knot 5_1 may have been tied as shown. We can see the knot shown here has the same number of crossings as the target knot. However, if we follow the knot round from the upper right end, we see it crosses over three strands, then under two, whereas the corresponding strand on the correct knot crosses over three, under one, then over one. Other than that, the rest of the crossings look correct. The one observed crossing change allows the knot to be deformed through Reidemeister moves, leaving us with knot 5_1, a very different looking knot to 8_21.





Knot 6.2

The knot 6_2 may have been tied as shown. Again, we can see the knot shown here has the same number of crossings as the target knot. Following the knot from the upper right end, we get a pattern of over, over, under, over for the crossings, instead of the target over, over, over, under. Following further on the strand, we see it is on the inside of the finished knot, as opposed to the outside of the target knot, resulting in further incorrect crossings. These changes allow the knot to be deformed through Reidemeister moves, leaving us with knot 6_2. The knot 6_2 is closer in number of crossings to 8_21 than 5_1 is, but analysing the changes that resulted in both knots being tied shows us fewer mistakes led to 5_1 than 6_2, so 5_1 can be seen as being closer to the intended 8_21.





The knot 5_2 may have been tied as shown. The knot looks very different to the knot 8_21 and does not have the same amount of crossings as the target picture. It is likely that when tying this knot, the participant lost track of where on the target picture they were, loosing track of crossings and resulting in a knot quite far from the intended.




More about Knots


The mathematical definition of a knot is an embedding of a circle into three-dimensional space. The simplest, or trivial, case of a knot is a simple loop (i.e. a circle). This is called an unknot. However, the involved topology of higher-order knots has fascinated mathematicians, chemists as well as physicists alike for centuries.


Formation of Knotted Structures in Chemistry


The formation of molecular knots is a common phenomenon in chemistry. Prominent examples include DNA and even certain proteins. Currently, we are investigating dynamical knot formation in excitable media, such as knots of concentrations in the Belousov-Zhabotinsky reaction- a fascinating, oscillating reaction (see picture on the right). We study both their mathematical modelling as well as experimental realisation. We are also interested in knot formation within supramolecular materials, like gels. Gels are formed from many smaller molecules, called gelators. We're interested to see how these gelators arrange themselves to form a gel and how they can knot together.




Light-Matter Interaction with Knotted Light


We study the physics of knotted light fields interacting with dilute atomic vapours. We are investigating ways to map the topologically complex properties of the light field onto the state of the atomic ensemble, both theoretically and experimentally.


Knots in Anthropology


We investigate the evolution of knot diversity observed in human material culture. Our aim is to develop a suitable mathematical framework to characterise variation in knots, and investigate the processes of innovation and cultural transmission that gave rise to them. The cultural evolution of knotting will be addressed through a combination of mathematical, statistical and phylogenetic modelling, in addition to experiments simulating the cultural transmission of knot tying.



Previous "Knots of the Month"


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.