026 027 A Link is what we call a collection of knots intertwined with each other, but not actually connected end-to-end. The simplest link is two unknots that don’t cross, called the Unlink. A collection of two knots is called a Hopf Link, and three knots are known as the Borromean Rings. Working with Knots The central problem of Knot Theory, known as the ‘Knot equivalence problem’, is determining whether two knots can be rearranged to be identical, and hence concluding that they are the same knot. Alan Turing, considered the father of modern computer science, wrote in his final publication, “A problem which might well be unsolvable is the one concerning knots”. Take the three knots on the left, for example, which can all be simplified to have less crossings. Can you guess which knot – one I’ve previously mentioned – they all represent? This example outlines the sheer difficulty of the knot equivalence problem. The Reidemeister moves and reducing knots The first step towards solving the knot equivalence problem was made in 1926, when Kurt Reidemeister proved that a specific order of just three simple moves can transform one knot to another if they are fundamentally the same. These are the Reidemeister moves: For example, have a go at a slightly easier example than the previous one. The first knot to the right is a projection of the trefoil knot, can you get it into its reduced form (the form where it has the least crossings), using the three Reidemeister moves stated above? Answer: Take top of the rope, do the following: Poke, Poke, Slide, Slide, Slide, Slide, Twist, Twist, Twist. If you can understand how we went from one knot to another, thereby proving their equivalence, you should have a go at reducing the other three knots on the previous page – however these require significantly more Reidemeister moves to fully reduce. Hint: they all reduce to the unknot. Tri-colourability and distinguishing knots The above moves are useful for finding if two simple knots are equivalent to each other, but what about more complex knots that would take forever to prove/disprove their equivalence? Tri-colourability refers to a technique that allows the undisputable disproof in the equivalence of two knots by colouring them in only three colours. Simply, if one knot is tri-colourable and the other is not, they are not the same knot. There are two rules to tri-colouring a knot: 1. You must use at least two colours 2. At crossings, the three intersecting strands must all be the same colour or different colours. Applications of Knot Theory – Materials development Knot Theory was historically seemingly useless pure maths, until 1989 when chemist Jean-Pierre Sauvage tied molecules around copper ions, forming the first ever synthetic knotted molecule. This trefoil knot gave the molecule new properties by trapping the atoms in higher energy states. Theoretically, this can be taken further by knotting a molecule into one of the currently known 159 billion knots, each producing an entirely new material. This field is very new for specific applications; chemists are just focused on creating molecular knots before thinking about materials development, though they hope to eventually build things like durable fabrics that are even stronger than Kevlar. So how do I tie my laces? Both of the common ways to tie your shoelaces include two trefoil knots on top of each other. If, after you tie the first knot, you tie the second one anti-clockwise around the loop, you will form the easily loosening granny knot. If, however, you tie the second knot clockwise around the loop, you will tie a reef knot, which – as widely known – is much stronger, and much more difficult to loosen. So, in the midst of early morning panic, remember to tie your laces in a reef knot – to save precious seconds later when your otherwise doomed knot would eventually untie itself. Peter Guthrie Tait, alongside one 3-crossing knot (Trefoil), one 4-crossing knot (Figure-8), two 5-crossing knots, three 6-crossing knots and seven 7-crossing knots. A Link is what we call a collec6on of knots intertwined with each other, but not actually connected end-to-end. The simpl st link is two unknots that don’t cross, called the Unlink. A collec6on of two knots is called a Hopf Link, and three knots are known as the Borromean Rings. orking with Knots Th central problem of Knot Theory, known as the ‘Knot equivalence problem’, is determining whether two knots can be rearranged to be ide 6cal, and hence concluding that they are the same knot. Alan Turing, considered the father of odern computer science, wrote in his final publica6on “A problem which might well be unsolvable is the one concerning knots”. Take the three knots on the le<, for example, which can all be simplified to have less crossings. Can you guess which knot – one I’ve previously men6oned – they all represent? This example outlines the shear difficulty of the knot equivalence problem. The Reidemeister move and reducing knots The first step towards s l ing t e knot quivalence problem was made in 1926, when Kurt Reidemeister proved that a specific order of just three simple moves can transform on knot to another if they are fundamentally the same. These are the Reidemeister moves: For example, have a go at a slightly easier example than the previous one. The first knot to the right is a projec<on of the trefoil knot, can you g t it into its r duced form (the form where it has t least crossings), using the three R i emeister moves stated above? Answer: Take top of the rope, do the following: Poke, Poke, Slide, Slide, Slide, Slide, Twist, Twist, Twist. If you can und rstand how we went from ne k ot to another, ther by proving their equivalence, you should have a go at reducing the other three knots on the previous page – however these require significantly more Reidemeister moves to fully reduce. Hint: they all reduce to the unknot. Tri-colourability and dis1nguishing knots The above moves are useful for finding if two simple knots are equivalent to each other, but what about more complex knots that would take forever to prove/disprove their equivalence? Twist Poke Slide crossings), using the three Reidemeister moves stated above? Answer: Take top of the rope, do the following: Poke, Poke, Slide, Slide, Slide, Slide, Twist, Twist, Twist. If you can understand how we went from one knot to another, thereby proving their equivalence, you should have a go at reducing the ot er thr e knots n the previous p ge – however these require significantly more Reidemeister moves to fully reduce. Hint: they all reduce t he unknot. Tri-colourability and dis1nguishing knots The above moves are useful for finding if two simple knots are equivalent to each other, but what about more complex knots that would take forever to prove/disprove their equivalence? Tri-colourability refers to a technique that allows the undisputable disproof in the equivalence of two knots by colouring them in only three colours. Simply, if one knot is tri-colourable and the other is not, they are not the same knot. There are two rules to tri-colouring a knot: 1. You must use at least two colours 2. At crossings, the three intersec6ng strands must all be the same colour or different colours. Applica+ons of Knot Theory – Materials development Knot Theory was historically seemingly useless pure maths, un6l 1989 when chemist Jean-Pierre Sauvage 6ed molecules around copper ions, f rming the first ever synthe6c knoIed molecule. This trefoil knot gave the molecule new proper6es by trapping he ato s in higher energy states. Theore6cally, this can be taken further by knojng a molecule into one of the currently known 159 billion knots, each producing an en6rely new material. This field is very new for specific applica6ons; chemists are just focused on crea6ng molecular knots before thinking about materials development, though they hope to eventually build things like durable fabrics that are even stronger than Kevlar. So how do I +e my laces? Both of the common ways to 6e your shoelaces include two trefoil knots on top of each other. If, a<er you 6e the first knot, you 6e the second one an6-clockwise around the loop, you will form the easily loosening granny knot. If, however, you 6e the second knot clockwise around the loop, you will 6e a reef knot, which – as widely known – is much stronger, and much more difficult to loosen. So, in the midst of early morning p nic, remember to 6e your laces in a reef knot – to save precious seconds later when your otherwise doomed knot would eventually un6e itself. The Reidemeister moves and reducing knots The first step towards solving the knot equivalence problem was made in 1926, when Kurt Reidemeister proved that a specific order of jus three simple mov c n transform one knot to another if they are fundamentally the same. These are the Reidemeister moves: For example, have a go at a slightly easier example than the previous one. The first knot to the right is a projec<on of the trefoil knot, can you get it into its reduced form (the form where it has the least crossings), using the three Reidemeister moves stated above? Answer: Take top of the rope, do the following: Poke, Poke, Slide, Slide, Slide, Slide, Twist, Twist, Twist. If you can understand how we went from one knot to another, thereby proving their equivalence, you should have a go at reducing the other three knots on the previous page – however these require significantly more Reidemeister moves to fully reduce. Hint: they all reduce to the unknot. Tri-colourability and dis1nguishing knots The above moves are useful for finding if two simple knots are equivalent to each other, but what about more complex knots that would take forever to prove/disprove their equivalence? Twist Poke Slide
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