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Notation and Basic Facts in Knot Theory

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Notation and Basic Facts in Knot Theory Appendix A Notation and Basic Facts in Knot Theory In this appendix, our aim is to provide a quick review of basic terminology and some facts in knot theory, which we use or need in this book. This appendix lists them item by item (without proof); so we do no attempt to give a full rigorous treatments; Instead, we work somewhat intuitively. The reader with interest in the details could consult the textbooks [BZ, R, Lic, KawBook]. • First, we fix notation on the circle S1 := {(x, y) ∈ R2 | x2 + y2 = 1}, and consider a finite disjoint union S1. A link is a C∞-embedding of L :S1 → S3 in the 3-sphere. We denote often by #L the number of the disjoint union, and denote the image Im(L) by only L for short. If #L is 1, L is usually called a knot, and is written K instead. This book discusses embeddings together with orientation. For an oriented link L, we denote by −L the link with its orientation reversed, and by L∗ the mirror image of L. • (Notations of link components). Given a link L :S1 → S3, let us fix an open tubular neighborhood νL ⊂ S3. Throughout this book, we denote the complement S3 \ νL by S3 \ L for short. Since we mainly discuss isotopy classes of S3 \ L,we may ignore the choice of νL. 2 • For example, for integers s, t ∈ Z , the torus link Ts,t (of type (s, t)) is defined by 3 2 s t 2 2 S Ts,t := (z,w)∈ C z + w = 0, |z| +|w| = 1 . This Ts,t is a knot if and only if s and t are relatively prime. • In addition, as a trivial example, the unknot (or the trivial knot) is the embedding K : S1 → S3; (x, y) → (x, y, z,w).If K is the unknot, the complementary space S3 K is homeomorphic to S1 × D2. • Two links L and L are called isotopic, if there is a smooth family of diffeomor- 3 3 phisms ht : S → S such that ht (L) is a link, and h1(L) = L and h0 = idS3 . © The Author(s) 2017 99 T. Nosaka, Quandles and Topological Pairs, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-981-10-6793-8 100 Appendix A: Notation and Basic Facts in Knot Theory Fig. A.1 The trefoil knot, figure eight knot, and the Tm,n-torus link with labeled arcs An invariant (of links) is a map I from the set of links to some set S such that, for any two links L and L which are isotopic, I (L) = I (L ) holds in S (Fig. A.1). 3 • The fundamental group π1(S K ) is sometimes called the knot group. • We will mention meridians and longitude. Given a link L, fix a tubular neighbor- hood νL ⊂ S3,as before. Then νL \ L is homotopic to #L S1 × S1, which is regarded as a S1-bundle over Im(L). Thus, we have a section s : Im(L) =#L S1 → νL \ L, called a framing. The -th component of Im(s) is called the (-th) longitude (of L). Furthermore, a representative circle of the -th component of the S1-fiber is called the (-th ) meridian. We denote them by l and m, respectively; see Fig. 2.4 as an example. Furthermore, with a choice of basepoint in the complementary space S3 L,we often regard the loops m and l, up to homotopy, as elements in the fundamental 3 group π1(S L). The abelian subgroup, P, generated by m and l is usually called the (-th) peripheral subgroup. Since we deal with links up to isotopy, we should give a remark: That is, in contrast to many choices of basepoints and the embedding 3 L, the peripheral subgroups P1,...,P#L ⊂ π1(S L) considered up to conjugacy depend on only the isotopy class of L. Example A.1 Fix four integers (a, b, s, t) ∈ Z4 with as + bt = 1. Then, it is known [BZ, Lic] that the knot group of the (s, t)-torus knot has the presentation 3 ∼ s t π1(S Ts,t ) = x, y, | x = y . Then, the meridian m and the longitude l are represented by xb ya and m−st xs , respec- s tively, and this π1 has the center generated by x . It is known [Sim] that a knot group 3 π1(S K ) contains a non-trivial center, if and only if K is a torus knot. • A(-th) Seifert surface of an oriented link L is an oriented surface embedded in 3 S L such that the boundary is the longitude l compatible with orientation. For any link and , we can construct a (-th) Seifert surface of L (via “Seifert algorithm”), although we can choose many such surfaces. • The integral homology of S3 L is as follows (which is immediately shown by the Alexander duality). Appendix A: Notation and Basic Facts in Knot Theory 101 ⎧ ⎪ Z, = , ⎨⎪ if i 0 Z#L , = , ( 3 ; Z) =∼ if i 1 Hi S L ⎪ Z#L−1, = , (A.1) ⎩⎪ if i 2 0, otherwise. 3 ∼ #L Here, generators of the first H1(S L) = Z are represented by the meridians m. We will briefly review geometrical properties of knots. First, we observe that the 3 isotopy types of knots are almost determined by the fundamental group π1(S K ) and a meridian-longitude pair. Precisely, Theorem A.2 (Waldhausen [Wal]) Let K and K be two knots in S3. Then, K is isotopic to either K or −K ∗ if and only if there is a group isomorphism Φ : 3 3 π1(S K ) −→ π1(S K ) such that Φ(m) = m and Φ(l) = l . 3 Furthermore, as is known, knot group π1(S K ) has interesting properties: Theorem A.3 (The loop theorem and the sphere theorem [Pa]) Let K be a knot. 3 1. The longitude l is not trivial in π1(S K ), if and only if K is not the unknot, 3 3 2. S K is an Eilenberg-MacLane space K (π, 1) with π = π1(S K ). That is, 3 the homotopy group πn(S K ) for any n ≥ 2 is zero. Furthermore, knot groups have a special property in the category of groups: Theorem A.4 (Gonzéalez-Acuña [Gon], Johnson [Joh]) Let G be a group, and fix z0 ∈ G. Then, the followings are equivalent: 3 1. There are a knot K with meridian m and a group epimorphism f : π1(S K ) → G such that f (m) = z0, −1 2. The group G is finitely generated, and is generated by the set {g z0g}g∈G . Proposition A.5 (Ryder [Ry]) Let Q K be the knot quandle of a knot K ; see Sect.2.3 ∼ 3 for the detailed definition: The map κ : Q K → As(Q K ) = π1(S K ) is injective, if and only if K is trivial or prime. Next, we review basic notions appearing in Theorem 7.20 (which are also impor- tant for the geometrization of knots). • A knot is said to be prime if, for any decomposition as a connected sum, one of the factors is unknotted. If not so, the knot is called composite. Here, see [Lic, BZ] for the definition of connected sum among (oriented) knots. • A knot K is cable,ifK is not any torus knot, and if there is a solid torus V embedded in S3 such that V contains K as the (p, q)-torus knot for some p, q ∈ Z. More generally, a knot K is satellite, if it is embedded in a small solid torus neighborhood of some knot K0, not the unknot, and K is not isotopic to K0 nor is contained in a ball inside the solid torus. Here, as is common in 3-dimensional topology, we should suppose that the unknot is not a torus knot, and torus knots are not cable knots. 102 Appendix A: Notation and Basic Facts in Knot Theory Theorem A.6 (JSJ-decomposition of knots (see [Bud, HWO] for the details)) Let 3 L ⊂ S be a prime knot. Then, there exists a sequence of open sets V1 ⊂ V2 ⊂···⊂ 3 Vn ⊂ S satisfying the following properties: 3 1. The set Vi is an open solid torus in S , and V1 contains the knot L. 2. The difference Vi+1 − Vi for any i ∈ Z≥0 is one of a hyperbolic link, or an 2 (ni , mi )-torus knot for some (ni , mi ) ∈ Z , or a composite knot in the solid torus. As is known, the decomposition is unique in some sense. Here, we should notice that L is cable iff Vj − Vj−1 for some j is an (n j , m j )-torus knot in the solid torus. Changing to 2-dimensional descriptions, we next focus on a diagrammatic approach to links: Precisely, we will review the Reidemeister Theorem A.7, which enables us to diagrammatically deal with links. To state, we define a link diagram to be a smooth immersion D :S1 → R2 whose double points are transverse, and in general position (so no triple intersections, or higher, occur) where the diagram contains the information if the arcs of the crossings are over or under. Here, an arc of D is a path as a subset of Im(D) running from a over-crossing to the next crossing. We call such a double point of D a crossing (in D). Two diagrams D and D are 2 2 called isotopic, if there is a smooth family of diffeomorphisms ht : R → R such that each ht (D) is a diagram, and h1(D) = D and h0 = idR2 .
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