![]() ![]() (This is a natural measure of complexity.) A minimal diagram of K is one with c(K) crossings. The crossing number c(K) of a knot K is the minimal number of crossings in any diagram of that knot. The Borromean rings are a 3-component example of a Brunnian link, which is a link such that deletion of any one component leaves the rest unlinked. The Borromean rings have the interesting property that removing any one component means the remaining two separate: the entanglement of the rings is dependent on all three components at the same time.Įxercise 1.2.6. Note that the individual components may or may not be unknots. A knot is therefore just a one-component link. Equivalence is defined in the obvious way. A link is simply a collection of (finitely-many) disjoint closed loops of string in R3 each loop is called a component of the link. See Adams’ “Knot book” for further historical information. Since then it has been “trendy” (this is a mixed blessing!) It even has some concrete applications in the study of enzymes acting on DNA strands. Knot theory was a respectable if not very dynamic branch of topology until the discovery of the Jones polynomial (1984) and its connections with physics (specifically, quantum field theory, via the work of Witten). It was not until Poincar´e had formalised the modern theory of topology around about 1900 that Reidemeister and Alexander (around about 1930) were able to make significant progress in knot theory. James Clerk Maxwell, William Thompson (Lord Kelvin) and Peter Tait (the Professor of maths at Edinburgh, and inventor of the dimples in a golf ball) began to think that “knotted vortex tubes” might provide an explanation of the periodic table Tait compiled some tables and gave names to many of the basic properties of knots, and so did Kirkman and Little. “Simplest” is clearly something we will need to define: how should one measure the complexity of knots? Although knots have a long history in Celtic and Islamic art, sailing etc., and were first studied mathematically by Gauss in the 1800s, it was not until the 1870s that there was a serious attempt to produce a knot table. Can one produce a table of the simplest knot types (a knot type means an equivalence class of knots, in other words a topological as opposed to geometrical knot: often we will simply call it “a knot”). One of the first tasks in the course will be to show that the trefoil is inequivalent to the unknot (i.e. We need to work much harder to prove this. On the other hand, wiggling a trefoil around for an hour or so and failing to make it look like the unknot is not a proof that they are distinct, merely inconclusive evidence. How might we prove inequivalence of knots? To show two knots are equivalent, we can simply try wiggling one of them until we succeed in making it look like the other: this is a proof. Mathematically, how do we go about formalising the definitions of knot and equivalence? Question 1.2.2. ![]() (Convince yourself of the latter using string or careful redrawing of pictures!)ġ.2. ![]() Any knot may be represented by many different diagrams, for example here are two pictures of the unknot and two of the figure-eight knot. Such two-dimensional representations are much easier to work with, but they are in a sense artificial knot theory is concerned primarily with three-dimensional topology. Actually the pictures above are knot diagrams, that is planar representations (projections) of the three-dimensional object, with additional information (over/under-crossing information) recorded by means of the breaks in the arcs. For example 51, 52 refer to the first and second of the two 5-crossing knots, but this ordering is completely arbitrary, being inherited from the earliest tables compiled. Some knots have historical or descriptive names, but most are referred to by their numbers in the standard tables. A knot is a closed loop of string in R3 two knots are equivalent (the symbol ∼ = is used) if one can be wiggled around, stretched, tangled and untangled until it coincides with the other. Motivation, basic definitions and questions This section just attempts to give an outline of what is ahead: the objects of study, the natural questions (and some of their answers), some of the basic definitions and properties, and many examples of knots. Lecture notes from Edinburgh course Maths 415. Reminder of the fundamental group and homotopy Date: March 15th 1999. Van Kampen’s theorem and knot groups 7.1. Alternating knots and the Jones polynomial 5. The Jones polynomial and its properties 4.5. A state-sum model for the Kauffman bracket 4.4. Formal definitions and Reidemeister moves 2.1. Motivation, basic definitions and questions 1.1. ![]()
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