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Knot for Knovices Knot for Knovices Tudor Ciurca∗, Kenny Lauy, Alex Cahillz, Derek Leungx Imperial College London 17 June 2019 Abstract This is a survey of some key results and constructions in knot theory, and is completed as part of the M2R group project. After developing some basic notions about knots and links, we will step into basic knot arithmetic and operations, exploring fundamental theorems such as Rei- demeister Theorem and Schubert's Theorem. Next, it would be a natural manner to step into the world of invariants. We will introduce various integer-valued and polynomial invariants, before wrapping up this chap- ter with Vassiliev's masterpiece of family of invariants. After that, we will explore Khovanov homology, which is the categorification of Jones' polynomial. It is indeed a powerful invariant, as we will see it solves the unknot recognition problem. Finally, we will switch our focus to the knot group and its Wirtinger presentation to end. ∗Email: [email protected] yEmail: [email protected], kc [email protected] zEmail: [email protected] xEmail: [email protected] 1 Contents 1 Introduction 3 1.1 What is a knot? . .3 1.2 When are two knots the same? . .4 1.3 Making finite and discrete models . .5 1.4 Projections . .8 1.5 Links . .9 1.6 Diagrams . 11 2 Reidemeister's Theorem and Seifert surfaces 13 2.1 Reidemeister's Theorem . 13 2.2 Basic knot arithmetic . 14 2.3 Seifert surfaces . 15 2.4 Schubert's Theorem . 18 3 Integer-valued and polynomial invariants 21 3.1 An overview . 21 3.2 3-colouring . 24 3.3 p-colouring . 30 3.4 Seifert matrix and Alexander polynomial . 32 3.5 Jones polynomial . 35 3.6 Vassiliev invariants . 44 4 From Jones' polynomial to Khovanov homology 48 4.1 Alternative version of Jones polynomial . 48 4.2 Khovanov homology . 49 4.3 Khovanov homology is an oriented link invariant . 55 4.4 Properties of Khovanov homology . 60 5 The knot group 65 5.1 Definition and basic properties . 65 5.2 Wirtinger presentation . 67 5.3 Meridian and longitude . 73 6 Bibliography 76 2 1 Introduction 1.1 What is a knot? Definition 1.1. An oriented knot is an injective continuous map f : S1 ! R3, where S1 := f(x; y) 2 R2 j x2 + y2 = 1g ⊂ R2 is the unit circle. Definition 1.2. A knot is a subset of R3 that is homeomorphic to S1. Examples include the unknot: 01 :(x; y) 7! (x; y; 0) and the trefoil knot: 31 : (cos t; sin t) 7! (r = 2 + cos(3t); θ = 2t; z = sin(3t)) which in Cartesian coordinates becomes: 3 2 3 3 31 :(x; y) 7! ((4x − 3x)(2x − 1); (4x − 3x + 2)(2xy); 3y − 4y ) Their projections are depicted below. (a) 01: unknot (b) 31: trefoil Figure 1: Projection of two knots Definition 1.3. If f is an oriented knot, then t 7! f(−t) is the orientation- reversal of the oriented knot. Definition 1.4. If f is an oriented knot, then the image of f is a knot, called the underlying knot. Remark 1.5. The underlying knot \forgets the orientation", since the orientation- reversal has the same underlying knot. 3 Since S3 := f(x; y; z; w) 2 R4 j x2 + y2 + z2 + w2 = 1g, the unit 3-sphere in R4, is the one-point compactification of R3, hence by noting that the image of a knot is already compact , one can instead define an oriented knot to be an injective continuous map f : S1 ! S3 without loss of generality, and similarly for a knot. No such maps can be surjective, for such a map would be bijective; but S1 is compact, S3 is Hausdorff, which implies that such a map would be a homeomorphism; but S1 and S3 are not homeomorphic, as one can remove two points from S3 and still get a connected space, yet the removal of any two points from S1 results in a disconnected space. Therefore, from an oriented knot S1 ! S3 we can pick a point not in the image, and then use stereographic projection on the complement of the point to make an oriented knot S1 ! R3; similarly for a knot. In the following we would identify S3 with R3 [ f1g freely, as well as switch the definition of an (oriented) knot freely between an injective continuous map S1 ! R3 and an injective continuous map S1 ! S3. 1.2 When are two knots the same? One would count \deformations" (to be defined) of a knot as the same knot, so we would need a notion of equivalence of knots that includes at least all the translations, non-zero dilations, and rotations. Deformations should be continuous, which leads to this attempted definition: Attempted definition: Two knots f; g : S1 ! R3 are equivalent if 1 3 there is a family of knots Fs : S ! R indexed by s 2 [0; 1] such 1 that F0 = f, F1 = g, and [0; 1] × S :(s; t) 7! Fs(t) is continuous. This definition is similar to the definition of a homotopy, but here we require 1 3 each intermediate function Fs : S ! R to be injective as well. However, one can check similarly that this equivalence is indeed an equivalence relation: • That f ∼ f is witnessed by Fs(t) := f(t). • Suppose f ∼ g is witnessed by Fs. Then g ∼ f is witnessed by F1−s. • Suppose f ∼ g is witnessed by Fs and g ∼ h is witnessed by Gs. Then 1 f ∼ h is witnessed by Hs where Hs := F2s for s 2 0; 2 and Hs := G2s−1 1 for s 2 2 ; 1 . It is continuous because F1 = G0 = g. The problem with this definition of equivalence, however, is that every knot is equivalent to the unknot under this definition. To see this, take any knot f : S1 ! R3. Since S1 is compact, there is t such that the x-coordinate of f(t) is maximum. Using that point as reference, break open the knot from that point while connecting the two resulting points with an arc growing in size to the right, then shrink everything from the left to the reference point, gradually re-parametrising the function so that the limit is still injective, and then finally use a translation to turn it into the standard unknot. The problem can be fixed by considering the whole space instead of just the knot: 4 Definition 1.6. Two oriented knots f; g : S1 ! R3 are equivalent if they 3 3 are ambient isotopic, i.e. there is a family of homeomorphisms Fs : R ! R for s 2 [0; 1] with F0(x) = x for all x and F1(f(t)) = g(t) for all t, such that 3 3 [0; 1] × R ! R :(s; x) 7! Fs(x) is continuous. 3 Definition 1.7. Two knots K1;K2 ⊂ R are equivalent if there is a family 3 3 of homeomorphisms Fs : R ! R for s 2 [0; 1] with F0(x) = x for all x and 3 3 F1(K1) = K2, such that [0; 1] × R ! R :(s; x) 7! Fs(x) is continuous. Remark 1.8. • These two notions of equivalence are both equivalence re- lations, using a similar argument. • If two oriented knots are equivalent, then clearly their underlying knots are also equivalent. • A version of the converse is true for nice enough knots, called \tame" knots, defined below in Definition 1.13. If two oriented knots are nice enough and their underlying knots are equivalent, then the two oriented knots are either equivalent, or one is equivalent to the orientation-reversal of the other (Theorem 1.16 below). 3 3 Given a continuous family of homeomorphisms Fs : R ! R where F0 is the identity, we say that F1 is isotopic to the identity. A homeomorphism R3 ! R3 is isotopic to the identity if and only if it is orientation-preserving (Theorem 3, Sanderson 1960), where a homeomorphism h : R3 ! R3 is said to 3 3 3 be orientation-preserving iff for every x 2 R the induced map h∗ : H3(R ; R − 3 3 fxg) ! H3(R ; R −ff(x)g) is the same map as the map induced by translation 3 hx(y) := y − x + h(x) for all x 2 R , where H•(A; B) is the relative singular homology. To facilitate computations, one notes that for any x 2 R3, if h is differentiable at x and the derivative at that point is non-singular, then h preserves orientation at x iff det(Dhx) > 0. This gives us a criterion that is simpler at times: Theorem 1.9. Two oriented knots f; g : S1 ! R3 are equivalent iff there is an orientation-preserving homeomorphism h : R3 ! R3 such that h ◦ f = g. 3 Similarly, two knots K1;K2 ⊂ R are equivalent iff there is an orientation- 3 3 preserving homeomorphism h : R ! R such that h(K1) = K2. 1.3 Making finite and discrete models To facilitate theory as well as actual computations, we need a finite and discrete version of knots. Unfortunately our current definition of knots is too broad which allows for some pathological examples called wild knots, which we do not prefer to compute with. Definition 1.10. A piecewise-linear knot is an oriented knot f : S1 ! R3 1 such that there are t1; ··· ; tn 2 S with tn = t1 such that f restricted to [ti; ti+1] is linear for every i = 1 ! n − 1. 5 Its underlying knot is called a polygonal knot, and is characterized by the 3 existence of p1; ··· ; pn 2 R with pn = p1 such that the knot is [p1; p2][[p2; p3][ ···[ [pn−1; pn] and that the line segments do not intersect except the fact that the end of [pi; pi+1] is the beginning of [pi+1; pi+2].
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