Namespaces
Variants
Actions

Difference between revisions of "Alexander invariants"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (convert png to latex)
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
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" />.
+
<!--
 +
a0113001.png
 +
$#A+1 = 172 n = 1
 +
$#C+1 = 172 : ~/encyclopedia/old_files/data/A011/A.0101300 Alexander invariants
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
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
+
{{TEX|auto}}
 +
{{TEX|done}}
  
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.
+
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} $.
  
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 projection of the manifold  $  \widetilde{M}  $
 +
onto the space  $  M $
 +
of orbits (cf. [[Orbit|Orbit]]) is a [[Covering|covering]] which corresponds to the kernel  $  K _ {a} $
 +
of the homomorphism  $  \gamma : G \rightarrow J _ {a} $
 +
of the [[Fundamental group|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|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|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
  
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
+
defines and is defined by the extension of modules  $  (**):  0 \rightarrow B _ {a} \rightarrow A _ {a} \rightarrow I _ {a} \rightarrow 0 $[[#References|[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 [[#References|[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|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.
  
<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>
+
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|loop]] representing  $  x $
 +
with all  $  k _ {i} $.
  
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" />.
+
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
  
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]].
+
$$
 +
\left (
 +
\frac{\partial  r _ {i} }{\partial  x _ {j} }
 +
\right ) ^ {\gamma _ {a} \phi } ,
 +
$$
 +
 
 +
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 [[#References|[4]]], [[#References|[8]]], [[#References|[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 [[#References|[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|Poincaré duality]] for the manifold  $  \widetilde{M}  $,
 +
taking into account the free action of  $  J  ^ {a} $[[#References|[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|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 [[#References|[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 [[#References|[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|Knot table]]). See also [[Knot theory|Knot theory]]; [[Alternating knots and links|Alternating knots and links]].
  
 
====References====
 
====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>
+
<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 $\nabla$-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>

Latest revision as of 06:42, 26 March 2023


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.

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 $\nabla$-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
How to Cite This Entry:
Alexander invariants. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Alexander_invariants&oldid=12088
This article was adapted from an original article by A.V. Chernavskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article