Alexander invariants

Invariants connected with the module structure of the one-dimensional homology of a manifold $ \widetilde{M} $, freely acted upon by a free Abelian group $ J ^ {a} $ of rank $ a $ with a fixed system of generators $ t _ {1} \dots t _ {a} $.

The projection of the manifold $ \widetilde{M} $ onto the space $ M $ of orbits (cf. Orbit) is a covering which corresponds to the kernel $ K _ {a} $ of the homomorphism $ \gamma : G \rightarrow J _ {a} $ of the fundamental group $ \pi _ {1} (M) = G $ of the manifold $ M $. Since $ K _ {a} = \pi _ {1} ( \widetilde{M} ) $, the group $ B _ {a} = K _ {a} / K _ {a} ^ \prime $, where $ K _ {a} ^ \prime $ is the commutator subgroup of the kernel $ K _ {a} $, is isomorphic to the one-dimensional homology group $ H _ {1} ( \widetilde{M} , \mathbf Z ) $. The extension $ 1 \rightarrow K _ {a} \rightarrow G \rightarrow J ^ {a} \rightarrow 1 $ generates the extension $ (*) : 1 \rightarrow B _ {a} \rightarrow G/ K _ {a} ^ \prime \rightarrow J ^ {a} \rightarrow 1 $, which determines on $ B _ {a} $ the structure of a module over the integer group ring $ \mathbf Z (J ^ {a} ) $ of the group $ J ^ {a} $( cf. Group algebra). The same structure is induced on $ B _ {a} $ by the given action of $ J ^ {a} $ on $ \widetilde{M} $. Fixation of the generators $ t _ {i} $ in $ J ^ {a} $ identifies $ \mathbf Z ( J ^ {a} ) $ with the ring $ L _ {a} = L _ {a} (t _ {1} \dots t _ {a} ) = \mathbf Z [ t _ {1} , t ^ {-1} \dots t _ {a} , t _ {a} ^ {-1} ] $ of Laurent polynomials in the variables $ t _ {i} $. Purely algebraically the extension

defines and is defined by the extension of modules $ (**): 0 \rightarrow B _ {a} \rightarrow A _ {a} \rightarrow I _ {a} \rightarrow 0 $[5]. Here $ I _ {a} $ is the kernel of the homomorphism $ \epsilon : L _ {a} \rightarrow \mathbf Z $ $ ( \epsilon t _ {i} = 1 ) $. The module $ A _ {a} $ is called the Alexander module of the covering $ \widetilde{M} \rightarrow M $. In the case first studied by J.W. Alexander [1] when $ M = M (k) $ is the complementary space of some link $ k $ of multiplicity $ \mu $ in the three-dimensional sphere $ S ^ {3} $, while the covering corresponds to the commutation homomorphism $ \gamma _ \mu : G(k) \rightarrow J ^ \mu $ of the link group, $ A _ \mu $ is the Alexander module of the link $ k $. The principal properties of $ G $ which are relevant to what follows are: $ G/ G ^ \prime $ is a free Abelian group, the defect of the group $ G $ is 1, $ G $ has the presentation $ \{ x _ {1} \dots x _ {m+1} ; r _ {1} \dots r _ {m} \} $ for which $ \gamma _ \mu (x _ {i} ) = t _ {i} $, $ 1 \leq i \leq \mu $; $ \gamma _ \nu (x _ {i} ) = 1 $, $ i > \mu $( cf. Knot and link diagrams). In the case of links the generators $ t _ {i} \in J ^ \mu $ correspond to the meridians of the components $ k _ {i} \subset k $ and are fixed by the orientations of these components and of the sphere.

As a rule, $ M $ is the complementary space $ M(k) $ of $ k $, consisting of $ \mu $ $ (n - 2) $- dimensional spheres $ k _ {i} $ in $ S ^ {n} $. In addition to the homomorphism $ \gamma _ {m} $, one also considers the homomorphism $ \gamma _ \sigma : G(k) \rightarrow J $, where $ \gamma (x) $ is equal to the sum of the link coefficients of the loop representing $ x $ with all $ k _ {i} $.

The matrix $ \mathfrak M _ {a} $ of the module relations of a module $ A _ {a} $ is called the Alexander covering matrix and, in the case of links, the Alexander link matrix. It may be obtained as the matrix

$$ \left ( \frac{\partial r _ {i} }{\partial x _ {j} } \right ) ^ {\gamma _ {a} \phi } , $$

where $ \{ x _ {i} ; r _ {i} \} $ is a presentation of the group $ G $. If $ \mu = 1 $, the matrix $ \mathfrak N _ {a} $ of module relations for $ B _ {a} $ is obtained from $ \mathfrak M _ {a} $ by discarding the zero column. The matrices $ \mathfrak M _ {a} $ and $ \mathfrak N _ {a} $ are defined by the modules $ A _ {a} $ and $ B _ {a} $ up to transformations corresponding to transitions to other presentations of the module. However, they can be used to calculate a number of module invariants. Alexander ideals are ideals of the module $ A _ {a} $, i.e. series of ideals $ E _ {i} ( A _ {a} ) $ of the ring $ L _ {a} : (0) \subseteq E _ {0} \subseteq \dots \subseteq E _ {i-1} \subseteq E _ {i} \subseteq \dots \subseteq (1) $, where $ E _ {i} $ is generated by the minors of $ \mathfrak M _ {a} $ of order $ (m - i) \times (m - i) $ and $ E _ {i} = L _ {a} $ for $ m - i < 1 $. The opposite numbering sequence may also be employed. Since $ L _ {a} $ is both a Gaussian ring and a Noetherian ring, each ideal $ E _ {i} $ lies in a minimal principal ideal $ ( \Delta _ {i} ) $; its generator $ \Delta _ {i} $ is defined up to unit divisors $ t _ {i} ^ {k} $. The Laurent polynomial $ \Delta _ {i} (t _ {1} \dots t _ \mu ) $ is simply called the Alexander polynomial of $ k $( or of the covering $ \widetilde{M} \rightarrow M $). If $ \Delta _ {i} \neq 0 $, it is multiplied by $ t _ {1} ^ {k _ {1} } \dots t _ \mu ^ {k _ \mu } $ so that $ \Delta _ {i} (0 \dots 0) \neq 0 $ and $ \neq \infty $. To the homomorphism $ \gamma _ \sigma $ there correspond a module $ \overline{A}\; $, ideals $ \overline{E}\; _ {i} $ and polynomials $ \overline \Delta \; _ {i} $, designated, respectively, as Alexander's reduced module, Alexander's reduced ideals and Alexander's reduced polynomials of $ k $( or of the covering $ {\widetilde{M} } _ \sigma \rightarrow M $). If $ \mu = 1 $, then $ A = \overline{A}\; $. $ \mathfrak M ( \overline{A}\; ) $ is obtained from $ \mathfrak M $ by replacing all $ t _ {i} $ by $ t $. If $ \mu \geq 2 $, $ \overline \Delta \; _ {1} $ is divisible by $ {(t - 1) } ^ {\mu - 2 } $. The polynomial $ \nabla (t) = {\overline \Delta \; } _ {1} (t) / {(t - 1) } ^ {\mu - 2 } $ is known as the Hosokawa polynomial. The module properties of $ A (k) $ have been studied [4], [8], [10]. The case of links has not yet been thoroughly investigated. For $ \mu = 1 $, the group $ H _ {1} ( \widetilde{M} ; R) $ is finitely generated over any ring $ R $ containing $ \mathbf Z $ in which $ \Delta (0) $ is invertible [7], in particular over the field of rational numbers, and, if $ \Delta (0) = +1 $, then also over $ \mathbf Z $. In such a case $ \Delta (t) $ is the characteristic polynomial of the transformation $ t : H _ {1} ( \widetilde{M} ; R) \rightarrow H _ {1} ( \widetilde{M} ; R) $. The degree of $ \Delta _ {1} (t) $ is equal to the rank of $ H _ {1} ( \widetilde{M} ; R) $; in particular, $ \Delta _ {1} (t) = 1 $ if and only if $ H _ {1} ( \widetilde{M} ; \mathbf Z) = 0 $. If $ n = 3 $, the link ideals have the following symmetry property: $ E _ {i} = {\overline{E}\; } _ {i} $, where the bar denotes that the image is taken under the automorphism generated by replacing all $ t _ {i} $ by $ t _ {i} ^ {-1} $. It follows that $ \Delta _ {i} ( t _ {1} ^ {-1} \dots t _ \mu ^ {-1} ) = t _ {1} ^ {N _ {1} } \dots t _ \mu ^ {N _ \mu } \Delta _ {i} ( t _ {1} \dots t _ \mu ) $ for certain integers $ N _ {i} $. This symmetry is the result of the Fox–Trotter duality for knot and link groups. It may also be deduced from the Poincaré duality for the manifold $ \widetilde{M} $, taking into account the free action of $ J ^ {a} $[3]. If $ \Delta _ {1} (t _ {1} \dots t _ \mu ) \neq 0 $, then the chain complex $ C _ {*} ( \widetilde{M} ) $ over the field of fractions $ P _ \mu $ of the ring $ L _ \mu $ is acyclic ( $ n = 3 $), and the Reidemeister torsion $ \tau \in P _ \mu / \Pi $ corresponding to the imbedding $ L _ \mu \subset P _ \mu $, where $ \Pi $ is the group of units of $ L _ \mu $, is defined accordingly. If $ \mu = 2 $, then $ \tau = \Delta _ {1} $; if $ \mu = 1 $, then $ \tau = \Delta _ {1} / t - 1 $( up to units of $ L _ \mu $). The symmetry of $ \Delta _ {1} $ for $ n = 3 $ is a consequence of the symmetry of $ \tau $. If $ \mu = 1 $, it follows from the symmetry of $ \Delta _ {i} (t) $ and from the property $ \Delta _ {i} (1) = \pm 1 $ that the degree of $ \Delta _ {i} (t) $ is even. The degree of $ \nabla (t) $ is also even [4]. The following properties of the knot polynomials $ \Delta _ {i} (t) $ are characteristic: $ \Delta _ {i} (1) = \pm 1 $; $ \Delta _ {i} (t) = t ^ {2k} \Delta _ {i} ( t ^ {-1} ) $; $ \Delta _ {i+1} $ divides $ \Delta _ {i} $; and $ \Delta _ {i} = 1 $ for all $ i $ greater than a certain value $ N $, i.e. for each selection $ \Delta _ {i} (t) $ with these properties there exists a knot $ k $ for which they serve as the Alexander polynomials. The Hosokawa polynomials [4] are characterized by the property $ \nabla (t) = t ^ {2k} \nabla (t ^ {-1} ) $ for any $ \mu \geq 2 $; the polynomials $ \Delta _ {1} $ of two-dimensional knots by the property $ \Delta _ {1} (1) = 1 $.

Alexander invariants, and in the first place the polynomials, are powerful tools for distinguishing knots and links. Thus, $ \Delta _ {1} $ fails to distinguish between only three pairs out of the knots in a table containing fewer than 9 double points (cf. Knot table). See also Knot theory; Alternating knots and links.

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