Lecture 3 Some Simple Knot Invariants In this lecture, we define several knot invariants, which take values in the natural numbers and have very simple definitions, but unfortunately are hopelessly hard to compute, and so are not of any practical use in knot theory. One of them, the “genus” of a knot, has a deep geometric meaning (related to oriented surfaces spanning the knot in 3-space), but is also difficult to calculate even for the simplest knots. We also introduce the notion of tricolorability, which does have a practical purpose, as it allows to prove that many knots are nontrivial. Unfortunately, it takes values in Z=2Z = fyes; nog and does help distinguishing knots. 3.1. Stick number By definition, the stick number of a knot is the least number of rectilinear edges needed to construct the given knot. It is obviously an ambient isotopy invariant of knots. Figure 3.1 shows that the stick number of the trefoil and the eight knot is ≤ 6. 1 1 7 4 5 4 6 2 2 3 5 6 3 Figure 3.1. The stick number of the trefoil and the eight knot is ≤ 6 Actually, the stick number of the trefoil is equal to 6; this can be proved by a tedious case-by-case argument. For knots more complicated than the trefoil, finding the exact value of the stick 1 2 number becomes hopelessly difficult. Thus the stick number may be a simple and nice invriant, but it is not practically useful for distiguishing nonequivalent knot diagrams. 3.2. Crossing number The crossing number of a knot is defined as the least number of crossings in any knot diagram representing the given knot. It is obviously an ambient isotopy invariant of knots. Figure 3.1 shows that the сrossing number of the trefoil is ≤ 3. Actually, the crossing number of the trefoil is equal to 3; this can be proved by showing that any knot with 2 crossings or less is trivial. However, for knots more complicated than the trefoil, finding the exact value of the crossing number becomes increasingly difficult. Figure 3.2 shows three knots with crossing numbers 5, 5, and 6. 51 52 61 Figure 3.2. The knots 51, 52, and 61 However, if the knot diagram is alternating and reduced, then the number of its crossings is automatically minimal – this is a recently proved classical conjecture that we accept without proof; a knot is called alternating if overpasses and underpasses alternate as we go around the knot and reduced if it has no crossings of the type shown in red in Fig.3.3. 3 Figure 3.2. A nonreducd knot 3.3. Unknotting number The unknotting number of a knot is defined as the least number of crossing changes needed to transfrom the given knot, presented as a knot diagram with a minimal number of crossings, into the trival one. The fact that any knot can be trivialized by crossing changes will be proved later in the course. It is easy to see that the trefoil and the eight knot have unknotting numbers equal to 1. For knots with a large number of crossings, finding the unknotting number is a tedious task and many different knots have the same unknotting number, so that this invariant is not of much use for distinguishing knots, but it is a rather curious invariant and has been seriously studied by knot theorists, especially those interested in the unknotting problem. 3.3. Tricolorability A knot is called tricolorable if it possesses a knot diagram whose “strands” can be colored in three colors so that in the vicinity of every crossing point either all three colors are present or only one 4 appears. Figure 3.3 shows that the trefoil is tricolorable while the eight knot isn’t. + + + + ! ! + Figure 3.3. Coloring the trefoil and the eight knot Theorem 3.1 Tricolorability is an invariant of knots, and any tricolorable knot is non trivial. Proof. The first assertion easily follows from the Reidemeister Lemma (1.1) – one easily checks that all three Reidemeister moves preserve tricolorability. The second assertion follows from the first and the obvious fact that the simplest diagram of the unknot (the round circle) is not tricolorable. The converse statement to Theorem 3.1 is not true, as Fig. 3.3 shows (the eight knot is non trivial but not tricolorable). 3.4. Digression about orientable surfaces At this stage the reader should know (or learn by reading this section) something about orientable surfaces. By definition, a surface (a.k.a. a topological two-dimensional manifold) is a compact topological space each point of which has a neighborhood homeo- morphic to the open disk. A surface is orientable if it does not contain a M¨obiusband. An example of a nonorientable surface is the Klein bottle. Orientable surfaces can be homeomorphically embedded in R3. By definition, a surface-with-boundary is a compact topological space each point of which has a neighborhood homeomorphic 5 either to the open disk or the open half disk 2 2 2 f(x; y) 2 R : x + y < 1; y ≥ 0g: The set of points of a surface-with-boundary N that do not have open disk neighborhoods is called the boundary of N and is denoted @N; it is not hard to show that @N consists of a finite number of circles. We will be interested in the case when the boundary consists of exactly one topological circle. A triangulation of a surface M is its representation in the form N [ 2 M = σi ; i=1 2 where the σi are (topological) triangles whose pairwise inter- sections are the empty set, or a common vertex, or a common side. A simple example of a triangulated sphere is the boundary of a tetrahedron. Other examples will be treated in the exercises. Fact 3.1. Any surface can be triangulated. Fact 3.2. The Euler characteristic χ(M) of a triangulated surface M, defined as χ(M): = V + E + F; where V is the number of vertices, E is the number of edges (sides)., and F is the number of faces (triangles) is a topological invariant of surfaces, i.e., homeomorphic surfaces have equal Euler characteristics. Fact 3.2. Any orientable surface is homeomorphic to exactly one of the following surfaces: the sphere S2, the torus T2, the sphere 2 2 with two handles M2, ... , the sphere with k handles Mk, ... These surfaces are classified by their Euler characteristic, which 6 equals 2, 0, -2, ... , 2 − 2k, ... respectively. They are shown in Fig.3.4. , , , … , ... , … 11111 2 2 2 2 2 kkkkk Figure 3.4. List of all orientable surfaces The genus g(M) of an oriented surface M is the number of its handles; thus the sphere S2 has zero handles, the torus T2, one 2 handle, Mn has n handles. It follows from Fact 3.3 that the genus of an oriented surface M is related to its Euler characteristic by the formula χ(M) = 2 − 2g(M): For a surface-with-boundary N with a single boundary component, we have χ(N) = 1 − 2g(N): 3.5. Seifert surface of a knot The Seifert surface of a knot is defined as an oriented surface in 3 R whose boundary is the given knot. Fig. 3.2 shows the knot 52 (a) and its Seifert surface (c), while (b) shows how the surface can be constructed. The reader can actually model this construction 7 by using paper colored in red and green on opposite faces, scissors, and Scotch tape. (a) (b) (c) Figure 3.5. Seifert surface for the 51 knot The next figure shows the construction of the Seifert surface of the trefoil, a paper model of which can also be easily made. 31 Figure 3.6. Seifert surface for the trefoil 3.6. The genus of a knot By definition the genus of a knot is the genus of its Seifert surface with the minimal number of handles. The genus of a knot is obviously a knot invariant. Once a Seifert surface S with the minimal number of handles of the given knot K has been constructed, the genus g(K) of the knot, which equals the genus g(S) of the surface by definition, can easily be calculated if one 8 knows its Euler characteristic χ(S) is, because the following for- mula holds: (1 − χ(S)) g(K) = g(S) = : 2 But we will not go into details here, because although the genus of a knot may be an interesting invariant, it not particularly useful for distinguishing (=classifying!) knots, just as all the other invariants mentioned in this lecture. 3.7. Exercises 3.1. Give a lower bound for the stick number of the knot 51. What is your conjecture for the value of the stick number of that knot? 3.2. What is your conjecture of the value of the stick number of the eight knot 41? Explain how would you go about proving it (without going into details)? 3.3. Prove that any knot diagram with two crossing points or less is trivial. 3.4. Prove that the only two knots with crossing number 3 are the two trefoils. 3.5. Find the unknotting number of the knots 51, 52, and 61. 3.6. Which of the knots 51, 52, and 61 are tricolorable? 3.7. On the classical model of the torus (square with identified opposite sides) draw a triangulation of the torus. 3.9. Indicate how the sides of a regular octagon can be identified so as to obtain a sphere with two handles.