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Knots by Lindsay Mooradian 1 1 Can you untie this Knot? Knot theory is the study of mathematical knots. A knot is a closed non-intersecting curve that is em- bedded in three dimensions. A break in a knot diagram represents a shadow, as knots are three dimen- sional objects being drawn on a two dimensional plane. The break helps us distinguish what strand is on top and what strand is below. Knot theorists are driven by the question: How do we tell if two knots are the same? 1 1 In a talk given by Candice Price at Harvey Mudd College in 2019, she discusses her research in knotted DNA structures. She uses well defined definitions and reasonings as to why knots are studied and the basics of knot theory. Candice Price (University of San Diego) on “Unravelling Biochemistry Mysteries: Knot theory applied to biochemistry” So, why do we study knots? Knot theory has real world applica- tions as simple as unknoting your headphones and as complex as understanding antibiotics and potentially curing cancer! 1 1 Price’s research involves knot theory because DNA can become knotted just like a mathematical knot. A topoisomerase I and II are proteins that are important for the replication transcription, unknotting, unlinking. When there is a knotted piece of DNA it needs to be untangled in order to replicate and make a new cell. This can cause cell death if the DNA can not become untangled. A type 11 to- poisomerase can come in when DNA is knotted and unlink these two pieces of DNA so it can replicate. Antibiotics can target the type 11s to not work so that those particular bad cells don’t replicate themselves and spread throughout the body. We can also target cancer cells, but when this happens we are telling topoisomerase 11s to not work at all, for good cells or bad cells. So, if we can advance our medicine to cure cancer it will be through knots and knot theory. Candice Price (University of San Diego) on “Unravelling Biochemisty Mysteries: Knot theory applied to biochemistry” We also study knots because of their strong history. One of the first records of knots being used in mathematics is in the Inca empire. 1 1 Quipu were recording knots used by many cultures from Andean South America pre-colonization. The knots were fashioned from cotton or camelid fiber strings. Quipu had numeric values and were used to count in the base ten system. Quipus were first noted around 1100 AD, but were replaced by European writing and numeral systems after the invasion of the Spanish Empire. D’altroy, Terence N. The Incas (2001) By the 1900s, knots in mathematics were gaining recognition again in mainstream academia. Peter Tait and John Newton Little published the first knot tabulation where they categorized knots up to ten crossings.1 1 Colberg writes about the beginnings of knot theory in academia. She emphasizes the very much diagram heavy upbringing of knot theory, as most mathematicians were first concerned with drawing and categorizing types of knots. The algebraic invariants and categorization came later on after many tables like the ones Tait and Little produced. Colberg, Erin. “A brief history of knot theory.” Página consultada a 8 (2017). The first book about knots, Knotentheorie was published by Kurt Reidemeister in 1932. 1His book was so revolutionary because he de- veloped the Reidemeister moves, which could be used to untangle a knot. 1 Kurt Reidemeister was born in Brunswick, Germany. After attending the University of Hamburg where he received his doctorate in Algebraic number theory he became a professor at the University of Konigsberg until he was dismissed by the Nazis for being politi- cally unsound. In 1927 he proved his three Reidemeister moves, which at the time was the most effective invariant. It is now the basis of knot theory and the first thing students of knot theory typically learn. Reidemeister, K. Knotentheorie. Julius Springer, 1932. We are able to look at the field of knot theory through knot di- agrams. In a knot diagram the string is a closed loop, as if you glued two ends of a shoelace together. If a knot can be untangled into a simple circle we categorize the knot as an “unknot”. If two knots are equivalent, their diagrams are related by a sequence of Reidemeister moves. There are three possible Reidemeister moves that we use to change a knot dia- gram. These moves change the diagram locally, leaving the rest of the diagram alone. A Reidemeister I move is defined by un- twisting or twisting a string. A Reidemeister II move is defined by moving one loop over another. A Reidemeister III move is defined by a slide, where a third string with can be moved over or under a crossing. Using these three moves we are now prepared to solve a problem in knot theory. First, we use a Reidemeister II move to untangle the crossing located in the dotted circle. We can see from the close up view of this section that it is a loop that we can pull under. Next, we use a Reidemeister II move again to undo the crossing in the middle of our knot. Looking at the close up version, we can see the loop. this time we pull it un- der in the opposite direction. Last, we are left with a circle with a twist in it. To un- twist the loop we use a Reidemeister I move. Now we are left with a complete circle--The unknot! Congratulations! You have just solved your first of many problems in knot theory! References Candice Price (University of San Diego) on “Unravelling Biochemistry Mysteries: Knot theory ap- plied to biochemistry” Colberg, Erin. “A brief history of knot theory.” Página consultada a 8, 2017. Epple, Moritz. “Knot Invariants in Vienna and Princeton during the 1920s: Epistemic Configura- tions of Mathematical Research.” Science in Context, 2004, www.maths.ed.ac.uk/~v1ranick/papers/ epple1.pdf. Goulding, Dominic. “Knot Theory: The Yang-Baxter Equation, Quantum Groups and Computation of the Homfly Polynomial.” Durham University, 2010 http://www.maths.dur.ac.uk/Ug/projects/ highlights/PR4/Goulding_Knot_Theory_report.pdf O’Conor, JJ. “Kurt Werner Friedrich Reidemeister.” Cardan Biography, 2014, www-history.mcs.st- and.ac.uk/history/Biographies/Reidemeister.html. Przytycki, Jozef H. “Classical Roots of Knot Theory.” Chnos, Solitons & Fracrals, 1998, https:// www.maths.ed.ac.uk/~v1ranick/papers/przytycki2.pdf Reidemeister, K. Knotentheorie. Julius Springer, 1932. “Reidemeister, Kurt Werner Friedrich.”. “Reidemeister, Kurt Werner Friedrich.” The Columbia Ency- clopedia, 6th Ed, Encyclopedia.com, 2019, www.encyclopedia.com/science/dictionaries-thesaurus- es-pictures-and-press-releases/reidemeister-kurt-werner-friedrich..
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