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A matrix associated with knots and links in order to investigate their topological properties by algebraic methods (cf. [[Knot theory|Knot theory]]). Named after H. Seifert [[#References|[1]]], who applied the construction to obtain algebraic invariants of one-dimensional knots in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s0838401.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s0838402.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s0838403.png" />-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s0838404.png" />-component [[Link|link]], i.e. a pair consisting of an oriented sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s0838405.png" /> and a differentiable or piecewise-linear oriented submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s0838406.png" /> of this sphere which is homeomorphic to the disconnected union of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s0838407.png" /> copies of the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s0838408.png" />. There exists a compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s0838409.png" />-dimensional orientable submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384011.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384012.png" />; it is known as the Seifert manifold of the link <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384013.png" />. The orientation of the Seifert manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384014.png" /> is determined by the orientation of its boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384015.png" />; since the orientation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384016.png" /> is fixed, the normal bundle to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384017.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384018.png" /> turns out to be oriented, so that one can speak of the field of positive normals to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384019.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384020.png" /> be a small displacement along this field, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384021.png" /> is the complement to an open tubular neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384022.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384023.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384024.png" /> is odd, one defines a pairing
+
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<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/s/s083/s083840/s08384025.png" /></td> </tr></table>
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associating with an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384026.png" /> the [[Linking coefficient|linking coefficient]] of the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384028.png" />. This <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384029.png" /> is known as the Seifert pairing of the link <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384030.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384032.png" /> are of finite order, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384033.png" />. The following formula is valid:
+
A matrix associated with knots and links in order to investigate their topological properties by algebraic methods (cf. [[Knot theory|Knot theory]]). Named after H. Seifert [[#References|[1]]], who applied the construction to obtain algebraic invariants of one-dimensional knots in  $  S  ^ {3} $.
 +
Let  $  L = ( S  ^ {n+} 2 , l  ^ {n} ) $
 +
be an  $  n $-
 +
dimensional  $  m $-
 +
component [[Link|link]], i.e. a pair consisting of an oriented sphere  $  S  ^ {n+} 2 $
 +
and a differentiable or piecewise-linear oriented submanifold  $  l  ^ {n} $
 +
of this sphere which is homeomorphic to the disconnected union of  $  m $
 +
copies of the sphere  $  S  ^ {n} $.  
 +
There exists a compact  $  ( n+ 1) $-
 +
dimensional orientable submanifold  $  V $
 +
of  $  S  ^ {n+} 2 $
 +
such that  $  \partial  V = l $;
 +
it is known as the Seifert manifold of the link $  L $.  
 +
The orientation of the Seifert manifold  $  V $
 +
is determined by the orientation of its boundary  $  \partial  V = l $;
 +
since the orientation of  $  S  ^ {n+} 2 $
 +
is fixed, the normal bundle to  $  V $
 +
in  $  S  ^ {n+} 2 $
 +
turns out to be oriented, so that one can speak of the field of positive normals to  $  V $.  
 +
Let  $  i _ {+} : V \rightarrow Y $
 +
be a small displacement along this field, where  $  Y $
 +
is the complement to an open tubular neighbourhood of  $  V $
 +
in  $  S  ^ {n+} 2 $.  
 +
If  $  n = 2 q - 1 $
 +
is odd, one defines a pairing
  
<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/s/s083/s083840/s08384034.png" /></td> </tr></table>
+
$$
 +
\theta : H _ {q} V \otimes H _ {q} V  \rightarrow  \mathbf Z ,
 +
$$
  
where the right-hand side is the [[Intersection index (in homology)|intersection index (in homology)]] of the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384036.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384037.png" />.
+
associating with an element  $  z _ {1} \otimes z _ {2} $
 +
the [[Linking coefficient|linking coefficient]] of the classes $  z _ {1} \in H _ {q} V $
 +
and  $  i _ {+} * z _ {2} \in H _ {q} Y $.  
 +
This  $  \theta $
 +
is known as the Seifert pairing of the link  $  L $.  
 +
If  $  z _ {1} $
 +
and $  z _ {2} $
 +
are of finite order, then  $  \theta ( z _ {1} \otimes z _ {2} ) = 0 $.  
 +
The following formula is valid:
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384038.png" /> be a basis for the free part of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384039.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384040.png" />-matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384041.png" /> with integer entries is called the Seifert matrix of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384042.png" />. The Seifert matrix of any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384043.png" />-dimensional knot has the following property: The matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384044.png" /> is unimodular (cf. [[Unimodular matrix|Unimodular matrix]]), and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384045.png" /> the [[Signature|signature]] of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384046.png" /> is divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384047.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384048.png" /> is the transpose of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384049.png" />). Any square matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384050.png" /> with integer entries is the Seifert matrix of some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384051.png" />-dimensional knot if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384052.png" />, and the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384053.png" /> is unimodular.
+
$$
 +
\theta ( z _ {1} \otimes z _ {2} ) + ( - 1 ) ^ {q} \theta ( z _ {2} \otimes z _ {1} ) = z _ {1} \cdot z _ {2} ,
 +
$$
  
The Seifert matrix itself is not an invariant of the link <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384054.png" />; the reason is that the construction of the Seifert manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384055.png" /> and the choice of the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384056.png" /> are not unique. Matrices of the form
+
where the right-hand side is the [[Intersection index (in homology)|intersection index (in homology)]] of the classes  $  z _ {1} $
 +
and $  z _ {2} $
 +
on  $  V $.
  
<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/s/s083/s083840/s08384057.png" /></td> </tr></table>
+
Let  $  e _ {1} \dots e _ {k} $
 +
be a basis for the free part of the group  $  H _ {q} V $.
 +
The  $  ( k \times k ) $-
 +
matrix  $  A = \| \theta ( e _ {i} \otimes e _ {j} ) \| $
 +
with integer entries is called the Seifert matrix of  $  L $.
 +
The Seifert matrix of any  $  ( 2 q - 1 ) $-
 +
dimensional knot has the following property: The matrix  $  A = ( - 1 )  ^ {q} A  ^  \prime  $
 +
is unimodular (cf. [[Unimodular matrix|Unimodular matrix]]), and for  $  q = 2 $
 +
the [[Signature|signature]] of the matrix  $  A + A  ^  \prime  $
 +
is divisible by  $  16 $(
 +
$  A  ^  \prime  $
 +
is the transpose of  $  A $).  
 +
Any square matrix  $  A $
 +
with integer entries is the Seifert matrix of some  $  ( 2 q - 1 ) $-
 +
dimensional knot if  $  q \neq 2 $,
 +
and the matrix  $  A + ( - 1 )  ^ {q} A  ^  \prime  $
 +
is unimodular.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384058.png" /> is a row-vector and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384059.png" /> a column-vector, are known as elementary expansions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384060.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384061.png" /> itself is called an elementary reduction of its elementary expansions. Two square matrices are said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384063.png" />-equivalent if one can be derived from the other via elementary reductions, elementary expansions and unimodular congruences (i.e. transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384064.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384065.png" /> is a unimodular matrix). For higher-dimensional knots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384066.png" /> and one-dimensional links <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384067.png" /> the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384068.png" />-equivalence class of the Seifert matrix is an invariant of the type of the link <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384069.png" />. In case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384070.png" /> is a knot, the Seifert matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384071.png" /> uniquely determines a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384072.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384073.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384074.png" /> is an infinite cyclic covering of the complement of the knot. The polynomial matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384075.png" /> is the Alexander matrix (see [[Alexander invariants|Alexander invariants]]) of the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384076.png" />. The Seifert matrix also determines the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384077.png" />-dimensional homology and the linking coefficients in the cyclic coverings of the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384078.png" /> that ramify over the link.
+
The Seifert matrix itself is not an invariant of the link  $  L $;
 +
the reason is that the construction of the Seifert manifold  $  V $
 +
and the choice of the basis  $  e _ {1} \dots e _ {k} $
 +
are not unique. Matrices of the form
 +
 
 +
$$
 +
\left \|
 +
 
 +
where $  \alpha $
 +
is a row-vector and $  \beta $
 +
a column-vector, are known as elementary expansions of $  A $,  
 +
while $  A $
 +
itself is called an elementary reduction of its elementary expansions. Two square matrices are said to be $  S $-
 +
equivalent if one can be derived from the other via elementary reductions, elementary expansions and unimodular congruences (i.e. transformations $  A \rightarrow P  ^  \prime  A P $,  
 +
where $  P $
 +
is a unimodular matrix). For higher-dimensional knots $  ( m = 1 ) $
 +
and one-dimensional links $  ( n = 1 ) $
 +
the $  S $-
 +
equivalence class of the Seifert matrix is an invariant of the type of the link $  L $.  
 +
In case $  L $
 +
is a knot, the Seifert matrix $  A $
 +
uniquely determines a $  \mathbf Z [ t , t  ^ {-} 1 ] $-
 +
module $  H _ {q} \widetilde{X}  $,  
 +
where $  \widetilde{X}  $
 +
is an infinite cyclic covering of the complement of the knot. The polynomial matrix $  t A + ( - 1 )  ^ {q} A  ^  \prime  $
 +
is the Alexander matrix (see [[Alexander invariants|Alexander invariants]]) of the module $  H _ {q} \widetilde{X}  $.  
 +
The Seifert matrix also determines the $  q $-
 +
dimensional homology and the linking coefficients in the cyclic coverings of the sphere $  S  ^ {2q+} 1 $
 +
that ramify over the link.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Seifert,  "Ueber das Geschlecht von Knoten"  ''Math. Ann.'' , '''110'''  (1934)  pp. 571–592</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.H. Crowell,  R.H. Fox,  "Introduction to knot theory" , Ginn  (1963)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Levine,  "Polynomial invariants of knots of codimension two"  ''Ann. of Math.'' , '''84'''  (1966)  pp. 537–554</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J. Levine,  "An algebraic classification of some knots of codimension two"  ''Comment. Math. Helv.'' , '''45'''  (1970)  pp. 185–198</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Seifert,  "Ueber das Geschlecht von Knoten"  ''Math. Ann.'' , '''110'''  (1934)  pp. 571–592</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.H. Crowell,  R.H. Fox,  "Introduction to knot theory" , Ginn  (1963)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Levine,  "Polynomial invariants of knots of codimension two"  ''Ann. of Math.'' , '''84'''  (1966)  pp. 537–554</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J. Levine,  "An algebraic classification of some knots of codimension two"  ''Comment. Math. Helv.'' , '''45'''  (1970)  pp. 185–198</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
For a description of the Seifert manifold in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083840/s08384079.png" />, i.e. the Seifert surface of a link, cf. [[Knot and link diagrams|Knot and link diagrams]].
+
For a description of the Seifert manifold in the case $  n = 1 $,  
 +
i.e. the Seifert surface of a link, cf. [[Knot and link diagrams|Knot and link diagrams]].

Revision as of 08:13, 6 June 2020


A matrix associated with knots and links in order to investigate their topological properties by algebraic methods (cf. Knot theory). Named after H. Seifert [1], who applied the construction to obtain algebraic invariants of one-dimensional knots in $ S ^ {3} $. Let $ L = ( S ^ {n+} 2 , l ^ {n} ) $ be an $ n $- dimensional $ m $- component link, i.e. a pair consisting of an oriented sphere $ S ^ {n+} 2 $ and a differentiable or piecewise-linear oriented submanifold $ l ^ {n} $ of this sphere which is homeomorphic to the disconnected union of $ m $ copies of the sphere $ S ^ {n} $. There exists a compact $ ( n+ 1) $- dimensional orientable submanifold $ V $ of $ S ^ {n+} 2 $ such that $ \partial V = l $; it is known as the Seifert manifold of the link $ L $. The orientation of the Seifert manifold $ V $ is determined by the orientation of its boundary $ \partial V = l $; since the orientation of $ S ^ {n+} 2 $ is fixed, the normal bundle to $ V $ in $ S ^ {n+} 2 $ turns out to be oriented, so that one can speak of the field of positive normals to $ V $. Let $ i _ {+} : V \rightarrow Y $ be a small displacement along this field, where $ Y $ is the complement to an open tubular neighbourhood of $ V $ in $ S ^ {n+} 2 $. If $ n = 2 q - 1 $ is odd, one defines a pairing

$$ \theta : H _ {q} V \otimes H _ {q} V \rightarrow \mathbf Z , $$

associating with an element $ z _ {1} \otimes z _ {2} $ the linking coefficient of the classes $ z _ {1} \in H _ {q} V $ and $ i _ {+} * z _ {2} \in H _ {q} Y $. This $ \theta $ is known as the Seifert pairing of the link $ L $. If $ z _ {1} $ and $ z _ {2} $ are of finite order, then $ \theta ( z _ {1} \otimes z _ {2} ) = 0 $. The following formula is valid:

$$ \theta ( z _ {1} \otimes z _ {2} ) + ( - 1 ) ^ {q} \theta ( z _ {2} \otimes z _ {1} ) = z _ {1} \cdot z _ {2} , $$

where the right-hand side is the intersection index (in homology) of the classes $ z _ {1} $ and $ z _ {2} $ on $ V $.

Let $ e _ {1} \dots e _ {k} $ be a basis for the free part of the group $ H _ {q} V $. The $ ( k \times k ) $- matrix $ A = \| \theta ( e _ {i} \otimes e _ {j} ) \| $ with integer entries is called the Seifert matrix of $ L $. The Seifert matrix of any $ ( 2 q - 1 ) $- dimensional knot has the following property: The matrix $ A = ( - 1 ) ^ {q} A ^ \prime $ is unimodular (cf. Unimodular matrix), and for $ q = 2 $ the signature of the matrix $ A + A ^ \prime $ is divisible by $ 16 $( $ A ^ \prime $ is the transpose of $ A $). Any square matrix $ A $ with integer entries is the Seifert matrix of some $ ( 2 q - 1 ) $- dimensional knot if $ q \neq 2 $, and the matrix $ A + ( - 1 ) ^ {q} A ^ \prime $ is unimodular.

The Seifert matrix itself is not an invariant of the link $ L $; the reason is that the construction of the Seifert manifold $ V $ and the choice of the basis $ e _ {1} \dots e _ {k} $ are not unique. Matrices of the form

$$ \left \|

where $ \alpha $ is a row-vector and $ \beta $ a column-vector, are known as elementary expansions of $ A $, while $ A $ itself is called an elementary reduction of its elementary expansions. Two square matrices are said to be $ S $- equivalent if one can be derived from the other via elementary reductions, elementary expansions and unimodular congruences (i.e. transformations $ A \rightarrow P ^ \prime A P $, where $ P $ is a unimodular matrix). For higher-dimensional knots $ ( m = 1 ) $ and one-dimensional links $ ( n = 1 ) $ the $ S $- equivalence class of the Seifert matrix is an invariant of the type of the link $ L $. In case $ L $ is a knot, the Seifert matrix $ A $ uniquely determines a $ \mathbf Z [ t , t ^ {-} 1 ] $- module $ H _ {q} \widetilde{X} $, where $ \widetilde{X} $ is an infinite cyclic covering of the complement of the knot. The polynomial matrix $ t A + ( - 1 ) ^ {q} A ^ \prime $ is the Alexander matrix (see Alexander invariants) of the module $ H _ {q} \widetilde{X} $. The Seifert matrix also determines the $ q $- dimensional homology and the linking coefficients in the cyclic coverings of the sphere $ S ^ {2q+} 1 $ that ramify over the link.

References

[1] H. Seifert, "Ueber das Geschlecht von Knoten" Math. Ann. , 110 (1934) pp. 571–592
[2] R.H. Crowell, R.H. Fox, "Introduction to knot theory" , Ginn (1963)
[3] J. Levine, "Polynomial invariants of knots of codimension two" Ann. of Math. , 84 (1966) pp. 537–554
[4] J. Levine, "An algebraic classification of some knots of codimension two" Comment. Math. Helv. , 45 (1970) pp. 185–198

Comments

For a description of the Seifert manifold in the case $ n = 1 $, i.e. the Seifert surface of a link, cf. Knot and link diagrams.

How to Cite This Entry:
Seifert matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Seifert_matrix&oldid=48646
This article was adapted from an original article by M.Sh. Farber (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article