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Introduction to Knot Theory Introduction to knot theory Summary of the lecture by Gregor Schaumann 2016 [email protected] Fakultät für Mathematik, Universität Wien, Austria This is a short summary of the lecture “Introduction to knot the- ory”, held by the author in the summer term of 2016 at the university of Vienna. The summary does not replace a script of the lecture, it is aimed at giving an overview on the topics that were covered during the course. It provides you with references and some suggestions for further reading. 1 Introduction Knots are topological objects familiar from daily experience. Two natural ques- tions arise: (i) What is a good mathematical concept of a knot? When are two knots considered to be equivalent? (ii) Can one classify all possible knots? Can one distinguish knots? It turns out, that classifying knots is very hard, knot theory focusses mainly on tools to distinguish knots, so called knot invariants. In modern physics, knots arise as worldlines of quantum particles in a 3 dimen- sional topological quantum field theory. Thus, such a theory in particular leads to knot invariants. A mathematical definition of these are the so called quantum invariants of knots. A famous work of Witten [Wit89] gives a physical reason why the Jones polynomial should be considered as a quantum invariant. This was made mathematically precise in [RT91] and [Saw96]. Plan of the lecture We define knots, their diagrams and a reasonable notion of equivalence between knots. Then we describe some basic phenomena that we observe when applying the definition. Among those the notion of connected sum of knots. Considering first examples of knots we are lead to the use of knot tables which lists knots that are prime with respect to the operation of connected sum. Indeed, every knot has a unique prime decomposition in prime knots. This is proven using Seifert surfaces in the second part. In the third part we start with a systematic investigation of knot invariants. Here, the Reidemeister moves are very useful, since they describe precisely the equivalence relation on knots using only knot diagrams. With this result we consider several combinatorial invariants, the knot colourings and discuss their geometric meaning using the knot group. The Alexander polynomial is intro- duced and its relation to the knot colourings is discussed. Finally, an alternative definition of the Alexander polynomial using the Conway-Alexander skein rela- tion motivates the definition of the Jones polynomial via the Kauffman bracket. This is used to prove the Tait conjecture. We do not just want to list one interesting invariant after the other, but we seek relations between the invariants and want to see in what sense they fall into certain classes of invariants. As important class we define quantum invariants in three steps: First we develop a diagrammatic calculus, then define tangles and finally define quantum invariants as devices to apply the diagrammatic calculus to tangles. We thereby avoid the (proper) languague of categories and functors due 2 to time reasons, but instead use an explicit presentation of tangles via generators and relations. Finally, Vassiliev invariants provide another systematic treatment of knot in- variants. Its relation to quantum invariants constructed from Lie algebras is mentioned at the end of the course. 1 Phenomenology of knots As reference for this section, see [CDM12, Chapter 1]. 1.1 Knots, what are they? We first define knots using smooth manifolds and embeddings. Definition 1.1. A parametrized knot is an embedding φ : S1 → R3, where S1 = {(x, y) ∈ R2|x2 + y2 = 1} is the circle. We always the orientation of circle that is counter-clockwise. This gives also an orientation to every parametrized knot. A reparametrization φ0 of a parametrized knot φ is an orientation preserving diffeomorphism diffeomorphism f : S1 → S1 such that φ0 = φ ◦ f. Definition 1.2. An oriented knot is an equivalence class of parametrized knots under reparametrizations. The following operation captures the deformation of one knot into another. Definition 1.3. Let φ1 and φ2 be two parametrized knots. An isotopy F : φ1 → 1 3 φ2 is a smooth map F : S × [0, 1] → R such that 1 (i) F (x, 0) = φ1(x) for all x ∈ S , 1 (ii) F (x, 1) = φ2(x) for all x ∈ S , (iii) F (−, u): S1 → R3 is a parametrized knot for all u ∈ [0, 1]. Lemma 1.4. Isotopy is an equivalence relation. Definition 1.5. The set of isotopy classes of oriented knots is denoted K. Its elements are called knots. 3 1.2 Knot diagrams Definition 1.6. A knot diagram for an oriented knot φ is a plane P ⊂ R3 with a projection π : R3 → P , such that π has at most finitely many double points on the knot. On each double point, the knot has to intersect transversally. The double points are over- and undercrossings. A knot diagram is called (i) alternating, when over- and undercrossings alternate when travelling across the knot diagram. (ii) reducibe, when it becomes disconnected when removing a small neighbour- hood of a single crossing. 1.3 Operations on knots 1 1 1 3 3 Denote by σS1 : S → S the orientation reversal of S and by τR3 : R → R the reflection along some plane in R3. This defines two operations on oriented knots φ : S1 → R3: (i) orientation reversal φ 7→ φ ◦ σS1 , (ii) mirror φ 7→ τR3 ◦ φ. This is well defined on equivalence classes K ∈ K and defines operation K 7→ σ(K) = K∗ and τ(K) = K. σ and τ form an action of Z/2Z × Z/2Z on K, where the generators 1 × 0 and 0 × 1 act via σ and τ, respectively. (That this is an action means just that σ2 = id = τ 2 and σ ◦ τ = τ ◦ σ) Definition 1.7. A knot is called (i) totally symmetric if all of K, K∗, K and K∗ are different, (ii) invertible if K∗ = K, (iii) plus-chiral if K = K, (iv) minus-chiral if K = K∗, (v) fully symmetric, if K = K∗ = K = K∗. Further operation: Connected sum of knots K1#K2 is defined by joining the two knots together. Knot tables list knots up to the operations σ, τ and connected sum (just prime knots). Definition 1.8. A knot is prime, if it is not the connected sum of two non-trivial knots. 4 2 Knots and surfaces We discuss two interplays of knots and surfaces: First knots on the torus, then surfaces with a knot as boundary. See [Sul00] as introductory reference. 2.1 Torus knots Knots on a surface: Consider T 2 = S1 × S1 the torus. The (p,q)-torus knot for p, q two coprime integers is the embedding S1 → T 2 that winds p times around the meridian (first S1) of T 2 and q times around the longitude (second S1) of T 2. Lemma 2.1. Every embedding of S1 into T 2 is isotopic to a (p, q)-torus knot for some values of p and q. 2.2 Seifert surfaces See [Sul00] for reference. Definition 2.2. A Seifert surface for an oriented knot K is an embedded oriented surface F in R3 with the knot K as boundary, ∂F = K. The Seifert construction gives a recipe how to construct a Seifert surface for every knot. Definition 2.3. The genus g(K) of a knot K ∈ K is the minimal genus g(FK ) among all Seifert surfaces FK for K. Recall, that the genus of a surface counts the number of holes of the surface (the sphere has genus 0, the torus genus 1, etc.). Theorem 2.4. (i) The genus detects the unknot: For a knot K, g(K) = 0 if and only if K is the unknot. (ii) The genus is additive under connected sum: g(K1#K2) = g(K1) + g(K2). A consequence is: Theorem 2.5. Any knot can be written as the connected sum of prime knots. Further Reading 2.6. For more details on Seifert surfaces, also in higher di- mensions, see [Rol76] 3 Invariants of knots Definition 3.1. Let S be a set. A knot invariant ν with values in S is a function ν : K → S. Main principle: If for an invariant ν, ν(K1) 6= ν(K2), then the knots K1 and K2 are not equivalent. 5 3.1 First examples • The genus is an invariant g : K → N0, • The crossing number c : K → N0 is the minimal number of crossings among all diagrams for a given knot. • The unknotting number u : K → N0 is the minimal number of crossings that is required to change any diagram of a given knot into a diagram for the unknot. c(K) The relation u(K) ≤ 2 holds. 3.2 Reidemeister moves Given two knot diagrams, how can we tell whether they represent equivalent knots? First, they could be just deformations of one of the other without changing any crossing. This is formalized by the notion of ambient isotopy: Two knot diagrams D and D0 are called ambient isotopic, if there exists a smooth map F : R2 × [0, 1] → R2 with F (−, 0) = id and F (−, 1)(D) = D0. The Reidemeister moves ΩI, ΩII and ΩIII (f.e. [CDM12, Thm 1.3.1]) are three moves on a knot diagram, that change only a small part of the knot diagram. However, by repeated use of these moves one can pass between all diagrams for a given knot and all diagrams for all equivalent knots. In between might Theorem 3.2. Two knots are equivalent if and only if two of their diagrams are related by a finite sequence of Reidemeister moves and ambient isotopies.
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