## Highlights - Thus knots form a semigroup under connect-sum\. In this semigroup, just as in the postive integers under multiplication, there is a notion of prime factorisation, which we will study later\. (Justin Roberts 6) - Question 1\.4\.2\. Hard research problem \(nobody has any idea at present\): give an intrinsically three-dimensional definition of an alternating knot \(i\.e\. without mentioning diagrams\)! (Justin Roberts 7) - Question 1\.4\.4\. Is every alternating diagram minimal? In particular, does every non-trivial alternating diagram represent a non-trivial knot? The answer turns out to be \(with a minor qualification\) yes, as we will prove with the aid of the Jones polynomial \(this was only proved in 1985\)\. (Justin Roberts 7) - All we really need is some kind of function taking values in an ordered set, having the property that the set of knots with complexity less than a given value is a finite set\. (Justin Roberts 13) - If we also impose the condition x\(0\) = x\(1\) then the initial and final point are made to coincide, so we have a parametric representation of a closed loop, rather than just an arc\. If we require that the map s 7→ x\(s\) is injective on the interval [0, 1\), then we enforce that the curve does not intersect itself\. (Justin Roberts 14) - Definition 2\.2\.1\. If K is a knot in R 3 , its projection is π\(K\) ⊆ R 2 , where π is the projection along the z-axis onto the xy-plane\. The projection is said to be regular if the preimage of a point of π\(K\) consists of either one or two points of K, in the latter case neither being a vertex of K\. Clearly a knot has an irregular projection if it has any edges parallel to the z-axis, if it has three or more points lying above each other, or any vertex lying above or below another point of K; on the other hand, a regular projection of a knot consists of a polygonal circle drawn in the plane with only “transverse double points” as self-intersections\. (Justin Roberts 17) - Fact 2\.2\.5\. “Regular projections are generic”\. This means “knots which have regular projections form an open, dense set in the space of knots”\. (Justin Roberts 17) - Unfortunately any knot can be represented by infinitely many different diagrams, which makes it unclear just how much of the information one can read off from a diagram (Justin Roberts 18) - Theorem 2\.3\.1 \(Reidemeister’s theorem\)\. Two knots K, K ′ with diagrams D, D ′ are equivalent if and only if their diagrams are related by a finite sequence D = D0 , D1 , \. \. \. , Dn = D ′ of intermediate diagrams, such that each differs from its predecessor by one of the following three \(really four, but we tend to take the zeroth for granted\) Reidemeister moves\. (Justin Roberts 18) - Proof of Reidemeister’s theorem (Justin Roberts 19) - Exercise 2\.3\.10\. Show that the number of regions in a diagram of a knot equals the number of crossings plus 2\. (Justin Roberts 21) - \. For example, in the above \(not terribly realistic\) picture, there are two clumps A and B, and so we might believe there are precisely two distinct knots represented by diagrams with n or fewer crossings\. But this is not true! Diagrams in different clumps need not represent different knots! Being in different clumps means only that there is no sequence of Reidemeister moves which only involves diagrams with n or fewer crossings joining them; it does not rule out the possibility that they are connected by a chain of R-moves which does involve at some point a diagram with more than n crossings\. In the picture, this corresponds to the existence of a path P , going below the part of the pyramid we have constructed, joining A and B\. (Justin Roberts 23) - This really can happen\. Here is an example: a 7-crossing diagram of the unknot \(it’s quite easy to see that it’s an unknot\), which cannot be directly reduced by a R1 or R2, and cannot be R3-ed at all! (Justin Roberts 23) - Theorem 2\.4\.1 \(Coward & Lackenby, 2010\)\. If D and D ′ are two diagrams of the same knot, both having less than n crossings, then there exists a sequence of Reidemeister moves relating them of length less than R\(n\), where R\(n\) = 2 2 2 \. \. \. 2 n with the number of 2s in the stack being 10 1000,000n \) (Justin Roberts 24) - What can one say but “!!!!”? (Justin Roberts 24) - Example 3\.2\.7\. The number of components µ\(L\) of a link L is an invariant (Justin Roberts 27) - Example 3\.2\.6\. The crossing number c\(K\) is the minimal number of crossings occurring in any diagram of the knot K\. (Justin Roberts 27) - This is a basic dichotomy exhibited by the knot invariants one commonly encounters\. One type is easily computable but must be proved to be invariant\. Such invariants tend not to have a clear topological interpretation (Justin Roberts 27) - The other type is obviously invariant \(anything defined in terms of “the minimal number of \. \. \. ” tends to be of this form\) but very hard to compute\. (Justin Roberts 27) - The interplay between these two kinds of invariants, attempting to use “computable” invariants to deduce facts about “non-computable” ones, forms a large part of knot theory\. (Justin Roberts 27) - Two \(the linking number and τ \) (Justin Roberts 27) - Corollary 3\.2\.11\. For any knot K, u\(K\) ≤ c\(K\)/2\. (Justin Roberts 28) - Exercise 3\.2\.12\. Prove that unknotting number and crossing number are examples of subadditive invariants, satisfying i\(K1 #K2 \) ≤ i\(K1 \) + i\(K2 \)\. (Justin Roberts 28) - and crossing number are examples of subadditive invariants, satisfying i\(K1 #K2 \) ≤ i\(K1 \) + i\(K2 \)\. \(It has long been thought that both of these should be equalities, but nobody has ever been able to prove or find a counterexample for either statement!\) (Justin Roberts 28) - The simplest useful, computable knot invariant is the number of 3-colourings τ \(K\), which we will now study (Justin Roberts 29) - Theorem 3\.3\.4\. The number of 3-colourings is a link invariant τ \(L\)\. Proof\. (Justin Roberts 29) - Exercise 3\.3\.8\. Try computing for other knots in the tables\. Can you explain why the answer is always divisible by three? Can you explain why it is always a power of three? (Justin Roberts 30) - Deduce by using repeated connect-sums of trefoils that there are (Justin Roberts 30) - infinitely many distinct knots\. (Justin Roberts 31) - (Justin Roberts 33) - Theorem 3\.3\.20\. The unknotting number of a knot is bounded below by u\(K\) ≥ log 3 \(τ \(K\)\) − 1 (Justin Roberts 33) - However, there are still pairs of inequivalent knots K, K ′ which have equal p-colouring invariants for all (Justin Roberts 36) - ; a more meaningful definition of the Alexander polynomial can be given using standard tools of algebraic topology \(namely, homology theory and covering spaces\) (Justin Roberts 39) - How did these polynomials emerge in the example calculations above? Although we thought we were trying to calculate the nullity of a matrix whose entries were simply complex numbers – because we had in mind a previously fixed value for t ∈ C ∗ – it now appears that we were doing some kind of Gaussian elimination for matrices whose entries are polynomials in Z [t] – that is, with t viewed as an indeterminate – at least until the end, where we used a rather ad hoc determinant argument in which t was once again viewed as a fixed complex number\. Is there a general process for computing the nullities {nt } simultaneously so that we can see they correspond to the roots of a polynomial ∆\(K\)? (Justin Roberts 42) - It’s very important in learning maths to have a good feeling for what is right, what is wrong, what is plausible or implausible, and to be able to mentally try to validate or invalidate \(by looking for simple counterexamples\) statements you may be presented with \(even your own it’s always good to try to “knock down” things which you think are true, but are uncertain about\)\. (Justin Roberts 43) - Over the next five years, lots of work was done leading to vast generalisations of Jones’ construction, and many connections to the field of “quantum algebra”, a subject more or less initiated by Vladimir Drinfeld in 1986\. Because of this, the mathematics dealing with “things not unrelated to the Jones polynomial” is now known as quantum topology\. (Justin Roberts 46) - It was invented in 1984 by Vaughan Jones \(hence ∗ the symbol V \), who was working in a completely different area of mathematics – operator algebras (Justin Roberts 46) - a really convincing and direct way to define it and illuminate its “topological meaning” – was still lacking\. Finally in 1989 the physicist Edward Witten gave an equally miraculous explanation using the methods of quantum field theory \(applied not to the real 4-dimensional physical world, but to a simplified 3-dimensional of the kind generally referred to by physicists as a “toy model”\)\. Because Witten’s explanation relies on the notoriously non-rigorous Feynman path integral method from QFT, it cannot be viewed \(at present\) as a rigorous mathematical theorem; but it is unquestionably the most profound interpretation we have \(at present\) of what the Jones polynomial really is\. Jones, Drinfeld and Witten were all awarded Fields Medals in 1990, in large part for these works\. (Justin Roberts 46) - \(Witten’s introduction of QFT as a method of making amazing conjectures in geometry and topology is probably the most important overall influence on these topics in the last 20 years\. People often ask whether knot theory is related to the physics of string theory\. The answer is “not in the sense you might imagine”\. The basic objects of string theory are indeed 1-dimensional loops, rather than the traditional 0-dimensional particles, but they are moving in a space of dimension much bigger than 3 \(in fact typically 10 or 26\), so there is no interesting knotting possible\. The sense in which the subjects are both string theory and the theory of the Jones polynomial are examples of quantum field theories\. \) (Justin Roberts 46) - From a knot-theoretic point of view, the Jones polynomial is a wonderful thing\. It is extremely good at distinguishing knots – it seems to be much more powerful than the previously-known computable knot invariants\. It can distinguish knots from their mirror images, which relatively few of the previously-known invariants could do\. It can be used to prove the 100-year-old “Tait conjectures” about alternating knots (Justin Roberts 46) - ∗ The way new mathematical inventions are named can be quite amusing\. Suppose you invent a new knot invariant: it would be considered crassly immodest to say something like “I call this the Roberts polynomial”, though it’s perfectly acceptable to give it the symbol R \(or even better, ρ!\) But by far the best technique is to give the polynomial an unusably convoluted \(“binary recursive regular isotopy polynomial”\) or insufficiently specific name \(“the diagram polynomial”\) or just not to name it at all \(there are lots of papers called things like “A new polynomial invariant of links”\)\. All of these tricks will \(fingers crossed\) force everyone else to refer to the thing using your name, but preserve your own modesty! (Justin Roberts 46) - Definition 4\.1\.1\. The Kauffman bracket polynomial of an unoriented link diagram D is a Laurent polynomial hDi ∈ Z [A ±1 ], defined by the following recursive rules: \(0\)\. It is invariant under planar isotopy of diagrams\. \(I have numbered this rule “zero” because it’s the sort of rule that almost goes without saying, and in fact will barely be mentioned from now on\.\) \(1\)\. It satisfies the skein relation (Justin Roberts 47) - \(6\)\. The word skein \(”a loosely wound ball of wool”\) was introduced into knot theory by John Conway in 1970\. As you might have noticed, most words having to do with entanglement have already been used in knot theory, so I assume he took out his thesaurus\.\.\. (Justin Roberts 49) - These two formulae describe precisely the non-invariance of the bracket under the first Reidemeister move R1\. It might seem that failing to be invariant under R1 makes further consideration of the bracket worthless, but this is not so! Lemma 4\.1\.6\. The Kauffman bracket is invariant under R2 and R3\. (Justin Roberts 49) - Definition 4\.2\.1\. If D is an oriented link diagram, then the writhe w\(D\) is just the sum of the signs of all crossings of D\. \(It differs from the total linking number in the fact that the self-crossings do contribute here, and there is no overall factor of 1 \.\) (Justin Roberts 52) - Lemma 4\.2\.2\. The writhe of an oriented link diagram is invariant under R2, R3 but changes by ±1 under R1\. (Justin Roberts 52) - Definition 4\.3\.1\. A state s of a diagram D is an assignment of either +1 or −1 to each crossing\. Clearly a c-crossing diagram has 2 c states\. Given a state s on D, we may form a new diagram sD by resolving or splitting the crossings of D: this means replaci (Justin Roberts 53) - Proposition 4\.3\.4\. The Kauffman bracket can be expressed by the explicit “state-sum” formula hDi = s hD|si, where s runs over all states of D, and hD|si denotes the contribution of the state s, namely hD|si = A P s \(−A 2 − A −2 \) |s\(D\)|−1 \. (Justin Roberts 53)