Lecture 17: The Alexander Module II

Transcription

1 Lecture 17: The Alexander Module II Notes by Jonier Amaral Antunes March 22, 2016 Introduction In previous lectures we obtained knot invariants for a given knot K by studying the homology of the infinite cyclic cover YK of the knot complement on the 3-sphere X K. Recall that the Alexander Module A K of K is the homology H 1 (YK ) seen as a module over the ring Λ := Z[t 1, t]. Here, the action of t on H 1 (YK ) is induced by the deck transformation t Aut(YK ) that identifies contiguous copies of Y Σ, the complement of a genus-g Seifert surface Σ. We can find a presentation for this module by first defining the Seifert form s : H 1 (Σ) H 1 (Σ) Z which is a bilinear form given by s([a], [b]) = lk(a, b) = lk(a, b + ) where a, b are closed curves on Σ, and a ± = i ± (a) are the inclusions of a in Y Σ by the identifications i ± : Σ Y Σ of Σ with each one of its two copies in Y Σ. If we pick a basis {α 1,..., α 2g } of H 1 (Σ), then we might represent s via the Seifert matrix V with entries V jk = s(α j, α k ) Subsequently, the presentation for A K is given by the result: Theorem 0.1. Given a knot K and a Seifert matrix Σ for K, let V be a Seifert matrix associated to the Seifert form on H 1 (Σ). Then the Alexander Matrix A := V tv T is a presentation matrix for the Alexander module A K, meaning we have the exact sequence: A Λ Z H 1 (Σ) Λ Z H 1 (Y Σ ) A K 0 (0.1) 1

2 The exact sequence (0.1) means that A K has 2g generators with relations given by the rows of A. Notice that if we reverse the chosen orientation of Σ exchanging the inclusions i ± by i the resulting Seifert matrix would be the transpose V T, since lk(a, b) = lk(a, b + ). Consequently, A T = V T tv would be the corresponding presentation. In other words, given an Alexander Matrix A, then its transpose A T presentation to the Alexander module. also gives a In this lecture we will further investigate the Alexander presentation in examples and derive other useful knot invariants from the Alexander matrix. From simplicity of notation, from now on we will denote A K = H 1 (Y K ) stating, if necessary when the Λ-module structure is being implied. 1 Examples of Alexander Matrices In order to get more familiarity with the Alexander presentation, let us compute the Alexander matrix in a few examples. As usual, we should start by considering the trefoil, K = 3 1. showed that we have a Seifert matrix Previously, we associated to a minimal genus Seifert surface for the trefoil. Therefore the matrix A = t 1 1 t t 1 is an Alexander matrix for 3 1. In terms of generators β 1, β 2 for H 1 (Yk ) we have the relations (t 1)β 1 + β 2 = 0 tβ 1 + (t 1)β 2 = 0 Thus we can express β 2 = (1 t)β 1. Therefore, H 1 (Yk ) is generated as a Λ-module by a single generator, β 1, subject to the relation tβ 1 + (t 1)(1 t)β 1 = (t 2 t + 1)β 1 = 0 2

3 Therefore we have the Λ-module isomorphism H 1 (Y K ) Λ t 2 t + 1 (1.1) which was obtained before in a less systematic way. Now let us consider the case of fibered knots. A fiber knot K is one such that the knot complement has a fiber bundle structure Σ X K S 1 where the fiber Σ is a Seifert surface for K. That means we have a homeomorphism X K Σ [0, 1]/ where the equivalence relation identifies Σ {0} with Σ {1} by (x, 0) (ϕ(x), 1) for some monodromy ϕ Aut(Σ) that is determined up to conjugacy class by the homeomorphism type of X K. Figure 1: Knot complement for fibered knot. In particular, the complement of the Seifert surface Y Σ is homeomorphic to Σ [0, 1], thus the infinite cyclic cover of Y K is given by Y K Σ R and the deck transformation t Aut(Y K ) acts via t((x, s)) = (ϕ(x), s + 1). Therefore, the fundamental group of the surface complement π 1 (YK ) = π 1 (Σ) = F 2g is free in 2g generators, with g the genus of Σ. But π 1 (YK ) Γ (1) K is the commutator subgroup of the knot group Γ K. By the theorem about the structure of the commutator subgroup proved in a previous lecture we know that if the commutator 3

4 Figure 2: Y K for fibered knot. subgroup is finitely generated, it is free in 2g(K) generators, where g(k) is the genus of K. Hence, we conclude that the fiber Σ must be a minimal genus Seifert surface for K. Furthermore, consider the map ϕ : H 1 (Σ) H 1 (Σ) induced on homology by the monodromy ϕ. From the action t((x, s)) = (ϕ(x), s + 1) we see that if a is a closed curve in Σ, then under the inclusion H 1 (Σ) H 1 (Y K ) we obtain t ([a]) = ϕ ([a]) in H 1 (Y K ). Thus, the Alexander module H 1 (Y K ) has a presentation A Λ Z H 1 (Σ) Λ Z H 1 (Y Σ ) H 1 (YK ) 0 where A is defined on generators {α j } for H 1 (Σ) as A(α j ) = tα j ϕ (α j ) which can be simply written as A = t Id 2g 2g ϕ (1.2) 2 The Alexander Polynomials Now we will proceed with extracting information from the Alexander matrix. Definition 2.1. Suppose M is a module over a commutative ring R, with a presentation R n A R m M 0 given by a m n presentation matrix A, where n m. The r-th elementary ideal E r of M is the ideal of R generated by all (m r + 1) (m r + 1) minors of A. Even though Definition 2.1 made explicit use of A, the elementary ideals do not 4

5 depend on the Alexander matrix. In fact, the E r depend only on the isomorphism-type of M and are invariant under change of presentation. Notice that in the case n = m and r = 1 we have E 1 = det A. Definition 2.2. The r-th Alexander ideal of the oriented knot K is the r-th elementary ideal of the Λ-module H 1 (YK ). The r-th Alexander polynomial is a generator of the minimal principal ideal of Λ containing E r.the first Alexander polynomial is called the Alexander polynomial of K, K (t) := det A = det(v tv T ). The Alexander polynomial is unique up to multiplication by a unit ±t n in Λ. Before we consider some elementary examples, notice that if H 1 (Y K ) is isomorphic to Λ/ p(t) for some p(t) Λ then we have a presentation Λ A Λ Λ p(t) 0 where A is given by multiplication by p(t). In that case, K (t) := det A = p(t). In the case K is the unknot, Y K D 2 R has trivial homology, H 1 (Y K ) 0. Thus, as a Λ-module, Consequently, K (t) = 1. H 1 (Y K ) Λ 1 Similarly, for the trefoil K = 3 1, from (1.1) we have K (t) = t 2 t + 1 t 1 + t 1. (2.1) Now let us have a look at the figure-eight knot, K = 4 1. A genus g = 1 Seifert surface Σ is depicted in Figure 3. Figure 3: Seifert surface for

6 In Figure 3, the central component positively oriented should be interpreted as coming out of the plane. Therefore, we have two generators {α 1, α 2 } for the homology H 1 (Σ) with projections {α 1 +, α 2 + } into H 1 (Y \ Σ) represented in Figure 4 and Figure 5, respectively. Figure 4: α 1 and α 2. Figure 5: α + 1 and α + 2. The linking numbers for the links formed by the α i and α + j from Figure 6 as can be easily calculated lk(α 1, α + 1 ) = 1 lk(α 1, α + 2 ) = 1 lk(α 2, α + 1 ) = 0 lk(α 2, α + 2 ) = 1 Figure 6: α 1, α 2, α + 1 e α + 2 linked The corresponding Seifert and Alexander matrices are: V = 1 1 = A = V tv T = 1 + t t 1 t Hence, det A = (1 t)(t 1) + t = (t 2 3t + 1). Thus we have the Alexander 6

7 polynomial for the figure-eight knot K (t) = t 2 3t + 1 t 3 + t 1 (2.2) The knots for which we just obtained the Alexander polynomial were all examples of fibered knots. Notice that the resulting polynomials were monic. That fact is an immediate consequence of (1.2) and might be stated as the following lemma. Lemma 2.3. Suppose K is a fibered knot. Then the Alexander polynomial of K is the characteristic polynomial of the map induced by the monodromy ϕ : K = det(t Id 2g 2g ϕ ) = t 2g + which is a monic polynomial. As discussed before, the degree of the Alexander polynomial is not invariant since we consider K (t) up to multiplication by a factor t n. However, we can obtain a well-defined algebraic invariant of K (t) by taking the difference between the highest and lowest power of t appearing in any particular representation of K (t). For any Laurent polynomial p(t) = c m t m + + c c n t n with n, m 0 we define the breadth of p(t) to be the sum n + m. So far we could only estimate the knot genus g(k) by finding superior bounds given by the Seifert construction. As the next Lemma shows, the breadth of K (t) can be used to obtain a lower bound for g(k), which in turn makes it often possible to completely determine g(k). Lemma 2.4. For any knot K, 2g(K) breadth K (t) with the equality holding if K fibers. Proof. Let V be the Seifert matrix associated to a minimal genus Seifert surface. Then, A = V tv T is a 2g(K) 2g(K) matrix. Therefore, det A is a polynomial of degree at most 2g(K). If K fibers then det(t Id 2g 2g ϕ ) is a polynomial of degree 2g(K). Lemma 2.5. For any knot K, the Alexander polynomial is symmetric in t modulo a 7

8 unit ±t n Λ. That is: K (t) K (t 1 ) Proof. This is just a simple computation K (t) = det(v tv T ) = det(v T tv ) = ( t) n det(v t 1 V T ) = ( t) n K (t 1 ) Lemma 2.5 suggests a preferred form to write the Alexander polynomial in a way that emphasizes symmetry t t 1, as it was done for 3 1 and 4 1 in (2.1) and (2.2). This is the standard convention for presenting K (t). Now let us analyze an example of non-fibered knots. Let K n be the twisted double of the unknot with n twists. The diagram for the case n = 2 can be seen in Figure 7, when K 2 = 5 2. When the number of crossings 2n 1 is negative, the twists are done in the opposite direction as the one indicated by Figure 7, that is, over and undercrossings in the lower part of the diagram should be inverted. Thus, the knot K 0 is the unknot and K 1 = 3 1 is the trefoil. Figure 7: Seifert surface for 4 1. Generators {α 1, α 2 } for H 1 (Σ) and the corresponding {α + 1, α + 2 } are also indicated in Figure 7. The only linking number that depends on n is the one between α 2 and α + 2 since α + 2 gives n twists around α 2. It follows that lk(α 1, α + 1 ) = 1 lk(α 1, α + 2 ) = 0 lk(α 2, α + 1 ) = 1 lk(α 2, α + 2 ) = n 8

9 So we have V = consequently 1 0 = A = tv V T = 1 n t 1 1 t n(t 1) Kn (t) = det A = n(t 1) 2 + t = nt 2 + (1 2n)t + n For n = 0, ±1 we have K0 (t) = t K1 (t) = t 2 t + 1 K 1 (t) = t 2 + 3t 1 and for no other n, Kn (t) is monic. We can state this as the following corollary. Corollary 2.6. The knots K n for n 0, ±1 are nontrivial, distinct and non-fibered. From the relations given by the Alexander matrix we can obtain the Alexander module for K n the same way as it was done for the trefoil. The conclusion is that H 1 (Y K n ) has one generator β as a Λ-module and the relation (nt 2 +(1 2n)t+n)β = 0, therefore: H 1 (Y K n ) Λ nt 2 + (1 2n)t + n Consequently, whenever n 0, ±1 the elements {t 2, t 3, t 4... } cannot be expressed as a Z-linear combination of lower order powers of t and represent independent Z- generators of H 1 (Y K n ). This implies that H 1 (Y K n ) is not finitely generated over Z. In particular, going back to K = 5 2 we get an example of knot with non-finitely generated knot group. Corollary 2.7. Set K = 5 2. Then H 1 (YK ) is not finitely generated over Z. Therefore the commutator subgroup Γ (1) K π 1 (YK ) is also not finitely generated. References [1] W. Lickorish. An Introduction to Knot Theory, Springer (1997). [2] D. Rolfsen. Knots and Links, AMS Publishing (1976). [3] L. H. Kauffman. Knots and Physics, World Scientific (2001). 9

10 [4] K. Murasugi. Knot Theory and its Applications, Birkhäuser (1996). 10