Knots Unravelled: From String to Mathematics is a book on the mathematics of knots, intended for schoolchildren and other non-mathematicians. It was written by mathematician Meike Akveld and mathematics publisher Andrew Jobbings, and published in 2011 by Arbelos, Jobbings's firm.
The main problem studied in the book is the use of knot invariants to test whether a loop is knotted or distinguish knots from each other. [1] [2] It has seven short chapters, [3] separated by "interludes" providing examples including Celtic knots, knotted papercraft, neckties, ropework, torus knots, and a form of the trefoil knot that can only sit on a plane with two points in contact. [2] [4] [5] [6] Small exercises, called "tasks" and often involving practical experiments rather than mathematical calculation, are scattered throughout the book, with answers at the end. [3] [4] [6]
The first chapter is introductory, and the second describes knot diagrams and the Reidemeister moves that change one diagram to another without changing the underlying knot. The next three chapters discuss particular knot invariants. These begin with the crossing number of a knot, the minimum number of crossings in its diagrams. Chapter four discusses another invariant, the unknotting number, the minimum number of local changes to a diagram that can unknot a given knot, while also discussing chirality (the phenomenon of a knot being different from its mirror image) and composite knots. Chapter five covers tricolorability, an invariant defined by coloring the arcs of a diagram according to certain rules. Chapter six generalizes the problem from knots to links, systems of more than two loops that cannot be separated from each other. [2] [3] [6] The final chapter, necessarily more mathematical than the others, is on the Jones polynomial. [3] [4] [5] [6]
Other material in the book includes historical asides, pointers to research topics, many illustrations, and an appendix with a table of small knots. [1]
This book is unusual among books on knot theory, an advanced mathematical subject, in being written for laypeople and schoolchildren, with no equations and little calculation. [5] Knot theorist Scott Taylor describes it as "filled with delightful mathematical ideas", an ideal way to attract bored students to mathematics, [4] and Jeff Johannes describes it as "my new favourite for introducing knot theory to non-mathematicians". [5] However, reviewer Roger Fenn suggests that, for use in secondary-school mathematics classes, the section giving solutions to the tasks needs expansion. [7]
In the mathematical theory of knots, the unknot, not knot, or trivial knot, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a knot tied into it, unknotted. To a knot theorist, an unknot is any embedded topological circle in the 3-sphere that is ambient isotopic to a geometrically round circle, the standard unknot.
In topology, knot theory is the study of mathematical knots. 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 so it cannot be undone, the simplest knot being a ring. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, . Two mathematical knots are equivalent if one can be transformed into the other via a deformation of upon itself ; these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing it through itself.
In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some invariants are indeed numbers (algebraic), but invariants can range from the simple, such as a yes/no answer, to those as complex as a homology theory (for example, "a knot invariant is a rule that assigns to any knot K a quantity φ(K) such that if K and K' are equivalent then φ(K) = φ(K')."). Research on invariants is not only motivated by the basic problem of distinguishing one knot from another but also to understand fundamental properties of knots and their relations to other branches of mathematics. Knot invariants are thus used in knot classification, both in "enumeration" and "duplication removal".
A knot invariant is a quantity defined on the set of all knots, which takes the same value for any two equivalent knots. For example, a knot group is a knot invariant.
Typically a knot invariant is a combinatorial quantity defined on knot diagrams. Thus if two knot diagrams differ with respect to some knot invariant, they must represent different knots. However, as is generally the case with topological invariants, if two knot diagrams share the same values with respect to a [single] knot invariant, then we still cannot conclude that the knots are the same.
Skein relations are a mathematical tool used to study knots. A central question in the mathematical theory of knots is whether two knot diagrams represent the same knot. One way to answer the question is using knot polynomials, which are invariants of the knot. If two diagrams have different polynomials, they represent different knots. In general, the converse does not hold.
In mathematics, the Borromean rings are three simple closed curves in three-dimensional space that are topologically linked and cannot be separated from each other, but that break apart into two unknotted and unlinked loops when any one of the three is cut or removed. Most commonly, these rings are drawn as three circles in the plane, in the pattern of a Venn diagram, alternatingly crossing over and under each other at the points where they cross. Other triples of curves are said to form the Borromean rings as long as they are topologically equivalent to the curves depicted in this drawing.
In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable with integer coefficients.
In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923. In 1969, John Conway showed a version of this polynomial, now called the Alexander–Conway polynomial, could be computed using a skein relation, although its significance was not realized until the discovery of the Jones polynomial in 1984. Soon after Conway's reworking of the Alexander polynomial, it was realized that a similar skein relation was exhibited in Alexander's paper on his polynomial.
In the mathematical area of knot theory, a Reidemeister move is any of three local moves on a link diagram. Kurt Reidemeister (1927) and, independently, James Waddell Alexander and Garland Baird Briggs (1926), demonstrated that two knot diagrams belonging to the same knot, up to planar isotopy, can be related by a sequence of the three Reidemeister moves.
In knot theory, a knot or link diagram is alternating if the crossings alternate under, over, under, over, as one travels along each component of the link. A link is alternating if it has an alternating diagram.
In mathematical knot theory, the Hopf link is the simplest nontrivial link with more than one component. It consists of two circles linked together exactly once, and is named after Heinz Hopf.
In mathematics, Khovanov homology is an oriented link invariant that arises as the cohomology of a cochain complex. It may be regarded as a categorification of the Jones polynomial.
The Tait conjectures are three conjectures made by 19th-century mathematician Peter Guthrie Tait in his study of knots. The Tait conjectures involve concepts in knot theory such as alternating knots, chirality, and writhe. All of the Tait conjectures have been solved, the most recent being the Flyping conjecture.
In the mathematical field of knot theory, the tricolorability of a knot is the ability of a knot to be colored with three colors subject to certain rules. Tricolorability is an isotopy invariant, and hence can be used to distinguish between two different (non-isotopic) knots. In particular, since the unknot is not tricolorable, any tricolorable knot is necessarily nontrivial.
In the mathematical field of knot theory, the bridge number is an invariant of a knot defined as the minimal number of bridges required in all the possible bridge representations of a knot.
In the mathematical area of knot theory, the unknotting number of a knot is the minimum number of times the knot must be passed through itself to untie it. If a knot has unknotting number , then there exists a diagram of the knot which can be changed to unknot by switching crossings. The unknotting number of a knot is always less than half of its crossing number. This invariant was first defined by Hilmar Wendt in 1936.
The Classical Groups: Their Invariants and Representations is a mathematics book by Hermann Weyl (1939), which describes classical invariant theory in terms of representation theory. It is largely responsible for the revival of interest in invariant theory, which had been almost killed off by David Hilbert's solution of its main problems in the 1890s.
When Topology Meets Chemistry: A Topological Look At Molecular Chirality is a book in chemical graph theory on the graph-theoretic analysis of chirality in molecular structures. It was written by Erica Flapan, based on a series of lectures she gave in 1996 at the Institut Henri Poincaré, and was published in 2000 by the Cambridge University Press and Mathematical Association of America as the first volume in their shared Outlooks book series.
Meike Maria Elisabeth Akveld is a Swiss mathematician and textbook author, whose professional interests include knot theory, symplectic geometry, and mathematics education. She is a tenured senior scientist and lecturer in the mathematics and teacher education group in the Department of Mathematics at ETH Zurich. She is also the organizer of the Mathematical Kangaroo competitions in Switzerland, and president of the Association Kangourou sans Frontières, a French-based international society devoted to the popularization of mathematics.
Math on Trial: How Numbers Get Used and Abused in the Courtroom is a book on mathematical and statistical reasoning in legal argumentation, for a popular audience. It was written by American mathematician Leila Schneps and her daughter, French mathematics educator Coralie Colmez, and published in 2013 by Basic Books.