European Stars And Stripes (Newspaper) - August 16, 1986, Darmstadt, Hesse Untying the secrets of knots by James Gleick new York times series of surprising discoveries is helping the practitioners of one of the purest forms of mathematics the theory of knots to attack a fundamental problem of their discipline How to distinguish one knot from another. At the same time the advances Are providing a new kind of tool scientists studying twisting looping string like structures in chemistry and molecular biology. Among other things the mathematical understanding of knots seems to hold direct Promise scientists deciphering the Way strands of molecules organize themselves into the Complex structures of dam. A theorist s knot is a formalized standardized idealized version of the everyday object a closed Loop of one dimensional string that winds through three dimensional space. It is one of the quintessential shapes that nature take in the mind of a mathematician. Problems of naming classifying and understanding knots have proved extremely hard. Picture of dam clearly show knots with turns and Crossings made visible by a new process of Coaling the molecules to prepare them the Electron Microscope. The mathematical discoveries seem to provide a key to the Way one Structure changes to another in life s Basic processes of replication and recombination. This is a very unusual joining of mathematics and biology said Kiyoshi Mizuuchi of the National institutes of health. You gain very Strong information about enzymes that is otherwise impossible to above All chemists and molecular biologists Are realizing that important features be understood by adopting the knot theorist s View of reality. Where euclidean geometry thinks of structures As rigid a knot theorist thinks in terms of Complete flexibility. That is the essence of topology the rubber Sheet geometry of which knot theory is one Branch. Distances and measurements Are irrelevant. If a certain knot be Bent twisted stretched squeezed or otherwise deformed into another knot without being Cut or untied then the two Are equivalent. The most Basic topological principles be difficult to prove. For example deciding whether two knots drawn on paper or modelled in string Are the same or different turns out to be a penetrating Puzzle. The outline of the problem is familiar to anyone who has Ever untied a package or a knotted a skein of yarn but its essence has occupied mathematicians nearly a Century and now seems closer than Ever to the heart of Modem geometry. There Are lot of hidden difficulties in the subject said Morgen b. This Lethwaite a mathematician at polytechnic of the South Bank in London. The problems Are easy to state but quite difficult to solve. And the techniques have become very indeed the great issues in knot theory sometimes surface in the Guise of recreational brain teasers. For example suppose cheating is defined As passing one part of a closed knot through another part. How Many times do you have to cheat to turn a Given knot into the Unknow or Circle one knot that be drawn with just eight Crossings the answer has Only recently been shown to be two with the proof requiring 300 pages of dense analysis. For other simple knots the Unk notting number remains unknown. A running theme of such problems is that a solution May seem obvious up to All but the very last degree of certainty. Cheating once May seem inadequate to Unknow. Two knots May seam different and after hours of unt Wisting and a looping you May feel sure experimentally. But perhaps you have just not been patient or Clever enough. Neither the Eye nor the hand Tell sure. A mathematician needs a system. Where a Sailor classifies knots in terms of physical characteristics a mathematician needs a Well defined Way of listing patterns of loops and Crossings. As knots get More complicated the complexities multiply. The simplest possible knot known As the Trefoil is drawn with just three Crossings. It is unique save its Mirror image. Similarly there is just one knot with four Crossings and just five. But 165 distinct knots have 10 Crossings and the total through 13 Crossings the highest number which a Complete Catalon now exists is 12,965 knots. Since the first great cataloguing a of knots in the 19th Century by the scottish physicist Peter Guthrie Tail and the american mathematician . Little mathematicians have tried searching invariants or fundamental properties that Tell one knot from another. The perfect invariant would distinguish any pair of knots. Short of perfection some invariants do better than others. The newest breakthrough in knot theory has been the discovery of invariants of a particularly powerful kind capable of distinguishing knots were other invariants failed. Several groups of mathematicians independently worked out sets of rules that would let them take any knot and systematically turn it into an algebraic expression known As a polynomial a combination of numbers and variables. The polynomial serves As a kind of Label the knot. Unlike the knots themselves the polynomials be told apart just by looking. And if the polynomials two knots Are different then the knots Are different unfortunately if they Are the same the knots May or May not be the same. As in All topological problems the particular size or shape of a Loop is irrelevant. All that matter Are the direction of the Crossings Over or under and their arrangement in relation to the other Crossings. The first polynomial invariant and until recently the Only one was discovered in the 1920s. It distinguished Many knots but mathematicians could not predict when it would work and when it would not. In 1984, Vaughan . Jones of the University of California at Berkeley an expert not in knot theory but in algebra discovered a new polynomial invariant and since then a succession of other mathematicians have taken his discovery further. It s a very exciting astonishing development said Joan s. Birman a Barnard mathematician who is an authority on knots and their near relations braids. It s important knot theory and it s important in a bigger sense because it s a Bridge Between two very different areas of mathematics where people never imagined there was a knot theory. Mathematical theorist have catalogued 249 Dot Tinct knot that to drawn with luit Locroi Lugt or leu. Bui cataloguing knot it a Tricky Button. Move right Vine exam pet that were thought 75 Yean to be different meaning that no amount of pulling and twisting could to uniform one into another. The a Rodlan to Lutton Cut Ting it not allowed a new York lawyer named Kenneth a. Perko or. Overturned a key Tenet of know theory by Date overlong that drawing a and b Wen actually the tame knot new Dot Coverlet an helping tote Tuch problem apparently Limpl yet theoretically quite profound. One approach let mathematician Ute the arrangement of crot Tongi in a knot diagram to produce an algebraic formula that Tenet at a Label the knot it two Label an different the the knot Mutt be too. And its application to genetics the Rule of knot conceived by mathematician Wen found to apply to the behaviour of Dan a Well. Tit knotted piece of Dan right we photographed through an Electron Miro scope. But before they Taw the to trend molecular to Loilit Wen Able to predict i existence on the bail of a mathematical sequence of Ever Mon comp heated knot. The stars and stripes Page 15
