Alexander invariants

Invariants connected with the module structure of the one-dimensional homology of a manifold , freely acted upon by a free Abelian group of rank with a fixed system of generators .

The projection of the manifold onto the space of orbits (cf. Orbit) is a covering which corresponds to the kernel of the homomorphism of the fundamental group of the manifold . Since , the group , where is the commutator subgroup of the kernel , is isomorphic to the one-dimensional homology group . The extension generates the extension , which determines on the structure of a module over the integer group ring of the group (cf. Group algebra). The same structure is induced on by the given action of on . Fixation of the generators in identifies with the ring of Laurent polynomials in the variables . Purely algebraically the extension

defines and is defined by the extension of modules [5]. Here is the kernel of the homomorphism . The module is called the Alexander module of the covering . In the case first studied by J.W. Alexander [1] when is the complementary space of some link of multiplicity in the three-dimensional sphere , while the covering corresponds to the commutation homomorphism of the link group, is the Alexander module of the link . The principal properties of which are relevant to what follows are: is a free Abelian group, the defect of the group is 1, has the presentation for which , ; , (cf. Knot and link diagrams). In the case of links the generators correspond to the meridians of the components and are fixed by the orientations of these components and of the sphere.

As a rule, is the complementary space of , consisting of -dimensional spheres in . In addition to the homomorphism , one also considers the homomorphism , where is equal to the sum of the link coefficients of the loop representing with all .

The matrix of the module relations of a module is called the Alexander covering matrix and, in the case of links, the Alexander link matrix. It may be obtained as the matrix

where is a presentation of the group . If , the matrix of module relations for is obtained from by discarding the zero column. The matrices and are defined by the modules and 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 , i.e. series of ideals of the ring , where is generated by the minors of of order and for . The opposite numbering sequence may also be employed. Since is both a Gaussian ring and a Noetherian ring, each ideal lies in a minimal principal ideal ; its generator is defined up to unit divisors . The Laurent polynomial is simply called the Alexander polynomial of (or of the covering ). If , it is multiplied by so that and . To the homomorphism there correspond a module , ideals and polynomials , designated, respectively, as Alexander's reduced module, Alexander's reduced ideals and Alexander's reduced polynomials of (or of the covering ). If , then . is obtained from by replacing all by . If , is divisible by . The polynomial is known as the Hosokawa polynomial. The module properties of have been studied [4], [8], [10]. The case of links has not yet been thoroughly investigated. For , the group is finitely generated over any ring containing in which is invertible [7], in particular over the field of rational numbers, and, if , then also over . In such a case is the characteristic polynomial of the transformation . The degree of is equal to the rank of ; in particular, if and only if . If , the link ideals have the following symmetry property: , where the bar denotes that the image is taken under the automorphism generated by replacing all by . It follows that for certain integers . 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 , taking into account the free action of [3]. If , then the chain complex over the field of fractions of the ring is acyclic (), and the Reidemeister torsion corresponding to the imbedding , where is the group of units of , is defined accordingly. If , then ; if , then (up to units of ). The symmetry of for is a consequence of the symmetry of . If , it follows from the symmetry of and from the property that the degree of is even. The degree of is also even [4]. The following properties of the knot polynomials are characteristic: ; ; divides ; and for all greater than a certain value , i.e. for each selection with these properties there exists a knot for which they serve as the Alexander polynomials. The Hosokawa polynomials [4] are characterized by the property for any ; the polynomials of two-dimensional knots by the property .

Alexander invariants, and in the first place the polynomials, are powerful tools for distinguishing knots and links. Thus, 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.

References

[1]	J.W. Alexander, "Topological invariants of knots and links" Trans. Amer. Math. Soc. , 30 (1928) pp. 275–306
[2]	K. Reidemeister, "Knotentheorie" , Chelsea, reprint (1948)
[3]	R.C. Blanchfield, "Intersection theory of manifolds with operators with applications to knot theory" Ann. of Math. (2) , 65 : 2 (1957) pp. 340–356
[4]	F. Hosokawa, "On -polynomials of links" Osaka J. Math. , 10 (1958) pp. 273–282
[5]	R.H. Crowell, "Corresponding groups and module sequences" Nagoya Math. J. , 19 (1961) pp. 27–40
[6]	R.H. Crowell, R.H. Fox, "Introduction to knot theory" , Ginn (1963)
[7]	L.P. Neuwirth, "Knot groups" , Princeton Univ. Press (1965)
[8]	R.H. Crowell, "Torsion in link modules" J. Math. Mech. , 14 : 2 (1965) pp. 289–298
[9]	J. Levine, "A method for generating link polynomials" Amer. J. Math. , 89 (1967) pp. 69–84
[10]	J.W. Milnor, "Multidimensional knots" , Conference on the topology of manifolds , 13 , Boston (1968) pp. 115–133