View source for Alexander invariants

Invariants connected with the module structure of the one-dimensional homology of a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a0113001.png" />, freely acted upon by a free Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a0113002.png" /> of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a0113003.png" /> with a fixed system of generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a0113004.png" />.

The projection of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a0113005.png" /> onto the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a0113006.png" /> of orbits (cf. [[Orbit|Orbit]]) is a [[Covering|covering]] which corresponds to the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a0113007.png" /> of the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a0113008.png" /> of the [[Fundamental group|fundamental group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a0113009.png" /> of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130010.png" />. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130011.png" />, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130013.png" /> is the [[Commutator subgroup|commutator subgroup]] of the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130014.png" />, is isomorphic to the one-dimensional homology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130015.png" />. The extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130016.png" /> generates the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130017.png" />, which determines on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130018.png" /> the structure of a module over the integer group ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130019.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130020.png" /> (cf. [[Group algebra|Group algebra]]). The same structure is induced on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130021.png" /> by the given action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130022.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130023.png" />. Fixation of the generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130024.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130025.png" /> identifies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130026.png" /> with the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130027.png" /> of Laurent polynomials in the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130028.png" />. Purely algebraically the extension

defines and is defined by the extension of modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130029.png" /> [[#References|[5]]]. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130030.png" /> is the kernel of the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130031.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130032.png" />. The module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130033.png" /> is called the Alexander module of the covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130034.png" />. In the case first studied by J.W. Alexander [[#References|[1]]] when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130035.png" /> is the complementary space of some link <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130036.png" /> of multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130037.png" /> in the three-dimensional sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130038.png" />, while the covering corresponds to the commutation homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130039.png" /> of the link group, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130040.png" /> is the Alexander module of the link <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130041.png" />. The principal properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130042.png" /> which are relevant to what follows are: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130043.png" /> is a free Abelian group, the defect of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130044.png" /> is 1, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130045.png" /> has the presentation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130046.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130048.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130050.png" /> (cf. [[Knot and link diagrams|Knot and link diagrams]]). In the case of links the generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130051.png" /> correspond to the meridians of the components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130052.png" /> and are fixed by the orientations of these components and of the sphere.

As a rule, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130053.png" /> is the complementary space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130054.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130055.png" />, consisting of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130056.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130057.png" />-dimensional spheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130058.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130059.png" />. In addition to the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130060.png" />, one also considers the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130061.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130062.png" /> is equal to the sum of the link coefficients of the [[Loop|loop]] representing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130063.png" /> with all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130064.png" />.

The matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130065.png" /> of the module relations of a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130066.png" /> is called the Alexander covering matrix and, in the case of links, the Alexander link matrix. It may be obtained as the matrix

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130067.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130068.png" /> is a presentation of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130069.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130070.png" />, the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130071.png" /> of module relations for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130072.png" /> is obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130073.png" /> by discarding the zero column. The matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130075.png" /> are defined by the modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130076.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130077.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130078.png" />, i.e. series of ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130079.png" /> of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130080.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130081.png" /> is generated by the minors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130082.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130083.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130084.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130085.png" />. The opposite numbering sequence may also be employed. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130086.png" /> is both a Gaussian ring and a Noetherian ring, each ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130087.png" /> lies in a minimal principal ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130088.png" />; its generator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130089.png" /> is defined up to unit divisors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130090.png" />. The Laurent polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130091.png" /> is simply called the Alexander polynomial of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130092.png" /> (or of the covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130093.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130094.png" />, it is multiplied by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130095.png" /> so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130096.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130097.png" />. To the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130098.png" /> there correspond a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a01130099.png" />, ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300100.png" /> and polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300101.png" />, designated, respectively, as Alexander's reduced module, Alexander's reduced ideals and Alexander's reduced polynomials of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300102.png" /> (or of the covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300103.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300104.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300105.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300106.png" /> is obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300107.png" /> by replacing all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300108.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300109.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300110.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300111.png" /> is divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300112.png" />. The polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300113.png" /> is known as the Hosokawa polynomial. The module properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300114.png" /> have been studied [[#References|[4]]], [[#References|[8]]], [[#References|[10]]]. The case of links has not yet been thoroughly investigated. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300115.png" />, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300116.png" /> is finitely generated over any ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300117.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300118.png" /> in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300119.png" /> is invertible [[#References|[7]]], in particular over the field of rational numbers, and, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300120.png" />, then also over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300121.png" />. In such a case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300122.png" /> is the characteristic polynomial of the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300123.png" />. The degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300124.png" /> is equal to the rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300125.png" />; in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300126.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300127.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300128.png" />, the link ideals have the following symmetry property: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300129.png" />, where the bar denotes that the image is taken under the automorphism generated by replacing all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300130.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300131.png" />. It follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300132.png" /> for certain integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300133.png" />. This symmetry is the result of the Fox–Trotter duality for knot and link groups. It may also be deduced from the [[Poincaré duality|Poincaré duality]] for the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300134.png" />, taking into account the free action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300135.png" /> [[#References|[3]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300136.png" />, then the chain complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300137.png" /> over the field of fractions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300138.png" /> of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300139.png" /> is acyclic (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300140.png" />), and the [[Reidemeister torsion|Reidemeister torsion]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300141.png" /> corresponding to the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300142.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300143.png" /> is the group of units of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300144.png" />, is defined accordingly. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300145.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300146.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300147.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300148.png" /> (up to units of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300149.png" />). The symmetry of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300150.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300151.png" /> is a consequence of the symmetry of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300152.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300153.png" />, it follows from the symmetry of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300154.png" /> and from the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300155.png" /> that the degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300156.png" /> is even. The degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300157.png" /> is also even [[#References|[4]]]. The following properties of the knot polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300158.png" /> are characteristic: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300159.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300160.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300161.png" /> divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300162.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300163.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300164.png" /> greater than a certain value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300165.png" />, i.e. for each selection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300166.png" /> with these properties there exists a knot <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300167.png" /> for which they serve as the Alexander polynomials. The Hosokawa polynomials [[#References|[4]]] are characterized by the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300168.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300169.png" />; the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300170.png" /> of two-dimensional knots by the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300171.png" />.

Alexander invariants, and in the first place the polynomials, are powerful tools for distinguishing knots and links. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300172.png" /> fails to distinguish between only three pairs out of the knots in a table containing fewer than 9 double points (cf. [[Knot table|Knot table]]). See also [[Knot theory|Knot theory]]; [[Alternating knots and links|Alternating knots and links]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.W. Alexander,   "Topological invariants of knots and links"  ''Trans. Amer. Math. Soc.'' , '''30'''  (1928)  pp. 275–306</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K. Reidemeister,   "Knotentheorie" , Chelsea, reprint  (1948)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R.C. Blanchfield,   "Intersection theory of manifolds with operators with applications to knot theory"  ''Ann. of Math. (2)'' , '''65''' :  2  (1957)  pp. 340–356</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  F. Hosokawa,   "On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011300/a011300173.png" />-polynomials of links"  ''Osaka J. Math.'' , '''10'''  (1958)  pp. 273–282</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  R.H. Crowell,   "Corresponding groups and module sequences"  ''Nagoya Math. J.'' , '''19'''  (1961)  pp. 27–40</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  R.H. Crowell,   R.H. Fox,   "Introduction to knot theory" , Ginn  (1963)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  L.P. Neuwirth,   "Knot groups" , Princeton Univ. Press  (1965)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  R.H. Crowell,   "Torsion in link modules"  ''J. Math. Mech.'' , '''14''' :  2  (1965)  pp. 289–298</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  J. Levine,   "A method for generating link polynomials"  ''Amer. J. Math.'' , '''89'''  (1967)  pp. 69–84</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  J.W. Milnor,   "Multidimensional knots" , ''Conference on the topology of manifolds'' , '''13''' , Boston  (1968)  pp. 115–133</TD></TR></table>