Knots theory and linear algebra sam nelson in this lecture, we will. By the early 1900s, kelvin s theory had been proven wrong. In 1858, he invented the mirror galvanometer knot theory s odd origins the modern study of knots grew out an attempt by three 19thcentury scottish physicists to apply knot theory to fundamental questions about the universe daniel s. Kelvin s insight continues to inspire new mathematics and has led to persistence of the topic in the history of science. This is isotopic to any simple closed curve in a plane embedded into r3. The possibility of founding a theory of elastic solids and liquids on the dynamics of more closelypacked vortex atoms may be reasonably anticipated. Applications of knot theory, volume 66 american mathematical. A revolution in knot theory 10 november 2011 this knot has gauss code o1u2o3u1o2u3. This conjecture motivated peter tait, a physicist, to create the rst knot table in which he described and classi ed all knots upto 10 crossings.
In mathematical language, a knot is an embedding of a circle in 3dimensional euclidean space, r 3 \displaystyle \mathbb r 3. The failed vortexatoms theory of matter by kelvin and tait had a profound impact on mathematics and physics. Knot theory is the part of topology that studies knots and links. The basic question one asks in knot theory is, given two knots how to know if they are the same knot or not. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined together so that it cannot be undone, the simplest knot being a ring. Knot theory project gutenberg selfpublishing ebooks. Knot theory and its applications occidental college. The early, significant stimulus in knot theory would arrive later with sir william thomson lord kelvin. Lord kelvin and tait, inspired by experiments of helmholtz, theorized that atoms were knotted tubes of aether, distinguished by their.
In the 19th century, lord kelvin made the inspired. Scottish physics and knot theorys odd origins introduction. The most prominent of these was english physicist, sir william thomson lord kelvin 18241907. A guide to the basic ideas and applications of knot theory, knots takes us from lord kelvin s earlyand mistakenidea of using the knot to model the atom, almost a century and a half ago, to the central problem confronting knot theorists today.
A conjecture by lord kelvin that atoms were knots in the aether led to a renewed interest in the studies of knots. In the late 1800s, most scientists believed that the universe was filled with a mysterious substance called ether, and all matter was thought to be entangled in it. Kelvin thought that atoms might be knotted vortices of \either. Two mathematical knots are equivalent if one can be transformed into the other via a deform. R3, considered up to isotopy smooth continuous deformations. It may be remarked in connexion with this anticipation, that the mere title of rankines paper on molecular vortices, communicated to the royal society of edinburgh in 1849 and 1850, was a. Guthrie tait were responsible for modern knot theory. A brief history of knot theory although knots have been around for thousands of years, they have only been a particular fascination for mathematicians for a little over a century. By 1970, knot theory had become a welldeveloped area of topology 7. One publication, journal of knot theory and its rami.
A reidemeister move is an operation that can be performed on the diagram of a knot whithout altering the corresponding knot. He became lord kelvin, or more precisely, baron kelvin of largs, 26 years later. Knot theory is the embedding of one topological space into another topological space. Knots and numbers haynes miller june 26, 2007 science and engineering program for teachers science and engineering program for teachers smoke rings and the kelvin atom. This theory inspired the celebrated scottish physicistpeter taitto undertake an extensive study and tabulation of knots in collaboration with c. An international conference on knot theory and related topics was held at osaka as a satellite conference of icm kyoto in 1990. Taits work especially founded the branch of topology called knot theory, with j. Scottish physics and knot theorys odd origins university of south. Examples of embeddings specifying knots f 1 abuse of notation here speci es a circle the in nite non knot example we looked at yesterday fails to be a knot because the. The rst mathematical study of knots was done by alexander t.
However topologists continued to study the knot theory as an area of pure mathematics. We call two knots equivalent if you can move one jumbled piece of string to look exactly like the other without cutting it open. Origins of knot theory in 1867, lord kelvin conjectured that atoms were knotted tubes of ether and the variety of knots were thought to mirror the variety of chemical elements. Here, we propose a natural classification of knots based on their intrinsic. Alexander 17 developed a way to classify most knots with up to nine crossings. Kelvins theory proved to be wrong the michelsonmorley. A classic example of this is the development of knot theory. Vandermonde which included the topological properties of knots.
A knot k is isotopic to a knot k0 we write k w k0 if there is a continuous motion of r3 mapping k onto k0. One of the rst times knot theory appeared as a subject of scienti c study was in 1860. A very brief summary of the history of knot theory 1880s. Knot theory knot theory is the mathematical study of an idealized model of knots. The paper, on knots, was written by the scottish mathematician. This theory of the ether was thoroughly debunked by. Ams short course lecture notes introductory survey lectures. Lord kelvin and a scientist named peter tait set out to classify knots. A measure of knot complexity is the number of minimum crossings that must occur when a knot is viewed as a twodimensional projection 3.
Dan 2006, knot theorys odd origins pdf, american scientist. Kelvin proposed in 1867 56 57 58 that physical atoms were knotted vortex tubes. Two knots are regarded as the same if one can be smoothly deformed into the other. This theory inspired the celebrated scottish physicist peter tait to undertake an extensive study and tabulation of knots in collaboration with c. The proceeding of this conference was knots 90 by akio kawauchi, e. Lord kelvin william thomson, 18241907 claimed that atoms knots made of ether. It was believed that a substance called ether filled all of space. From these applied considerations knot theory became a vibrant branch of pure mathematics. The modern mathematical study of knots was started by lord kelvin in the late 19th century, who thought that atoms might be knots in the luminifereous ether, the hypothetical space lling substance thought to carry light waves. It primarily uses algebraic and geometric techniques to study topological objects. May 01, 1997 in 1867, lord kelvin proposed that atomsthen considered to be elementary particlescould be described as knotted vortex tubes in either1. Hydrogen would be one kind of knot, oxygen a different kind of knot and so forth throughout the.
Knot theory was rst proposed as the awed vortex model of atoms by lord kelvin, but was later proved to be incorrect. The knot group of a knot awith base point b2s3 ima is the fundamental group of the knot complement of a, with bas the base point. Three dimensions of knot coloring university of south alabama. This volume is a collection of research papers devoted to the study of relationships between knot theory and the foundations of mathematics, physics, chemistry, biology and psychology. Lord kelvins theory of the atom stated that chemical properties of elements were re. This classi cation was meant to aid in the classi cation of atoms. After seeing taits colliding smoke ring experiments, thompson conceived vortex atom theory, where the rst attempt of applied knot theory was seen. This closing of the string prevents you from untying the knot, or turning it into any other kind of knot, without first breaking the string. Pdf applications of knot theory download full ebooks online. Experiments and simulations reveal mechanical properties of knots. A knot is a circle smoothly embedded in 3dimensional euclidean space or its compacti cation, the 3sphere. He believed that the different elements may then be determined by the different possible knots. Included are reprints of the work of lord kelvin sir william thomson on the 19th century theory of vortex atoms, reprints of modern papers on knotted flux in. In topology, knot theory is the study of mathematical knots.
Lord kelvin, in an attempt to reconcile several competing atomic theories, proposed that atoms had a knotted structure. During the 1860s, the scientific world was divided into two groups. Unexpected ramifications of knot theory asia pacific math. Nov 10, 2011 in the 19th century, lord kelvin made the inspired guess that elements are knots in the ether. Mar 24, 2019 a very brief summary of the history of knot theory 1880s. An introduction to knot theory and the knot group 5 complement itself could be considered a knot invariant, albeit a very useless one on its own. Alexander, two topologist started studying knots from a knot group point of view and the rst. Pdf history of knot theory jozef przytycki academia. Tait 1876, a colleague of kelvin knots to 7 crossings kirkman 1885, british knot projections little 1885, nebraska knots to 10 crossings by 1900, tait, kirkman, little had produced all 10 crossing knots and all 11 crossing alternating knots.
Tait set out to systematically enumerate knots based on their crossing number. He also presents the remarkable applications of knot theory to. Inspired by lord kelvin s vortex theory of the atom 59, the scottish physicist p. Included are reprints of the work of lord kelvin sir. Knots and applications series on knots and everything. This honors thesis introduces some fundamental ideas of knot theory in a way. Informally, one thinks of a knot as any of the shapes that can be made by a. Knot theory course description take a piece of string, jumble it up, then seal the ends together. Some fundamental theorems of involved in knot theory include one involving reidemeister moves transforming knots into unknots and knot. Taits study on vortices inspired lord kelvin to look for a description of atoms in terms of knots in the aether and this, in turn, motivated. The recent resurgence of knot theory in physics can be thought of as.
In the late 1800s lord kelvin suggested that atoms might represent knots in the ether, with di erent elements corresponding to di erent types of knots. A and b colorchanging mechanoresponsive fibers confirm the stress patterns predicted by continuum simulations for the t refoil knot a and the figureofeight knot b during the tightening process movie s1. Stable knotlike structures in classical field theory nature. Nov 10, 2011 a planar knot diagram is a picture of a knot, drawn in a twodimensional plane, in which every point of the diagram represents at most two points in the knot. A mathematicians knot is the same as the one you might tie in a piece of string, but with the ends of the string connected together.
This meant that physicists, such as peter guthrie tait, tried to create a table of the elements corresponding to a classi. Thompson providing some early mathematical advancements. Knots have been used for basic purposes such as recording information, fastening and tying objects together, for thousands of years. But it was lord kelvins theory that atoms were knotted vortices in the ether that sparked serious interest in determining the possible knots. Applications of knot theory were rst seen in the nineteenth century with william thompson, also referred to as lord kelvin, and peter tait. Notice that you cant untie the knot because youve permanently sealed the ends together. Planar knot diagrams have long been used in mathematics as a way to represent and study knots. The principal objective of the knot theory is to provide a simple way of classifying and ordering all the knot types. After watching one of these experiments, william thomson, later lord kelvin. Gauss was interested in the magnetic eld produced by an electrical current owing through a knotted loop of wire. The discovery of the jones polynomial by vaughan jones in 1984 not only showed a connection between knot theory and di erent areas of mathemat. Lord kelvin who theorized in the 1880s that atoms were knotted or linked vortex. As a branch of topology, knot theory is a developing field, with many unresolved questions, including the ongoing search for an algo. Applied physics topological mechanics of knots and tangles.
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