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''knot cobordism (proper bordism of knots, see [[Bordism|Bordism]])''
 
''knot cobordism (proper bordism of knots, see [[Bordism|Bordism]])''
  
An equivalence relation on the set of knots, weaker than the isotopy type relation. Two smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c0228001.png" />-dimensional knots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c0228002.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c0228003.png" /> are said to be cobordant if there exists a smooth oriented <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c0228004.png" />-dimensional submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c0228005.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c0228006.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c0228007.png" /> is homeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c0228008.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c0228009.png" />. Here the minus sign indicates the opposite orientation. Knots cobordant to the trivial knot are called cobordant to zero, or slice knots. The set of (cobordance) equivalence classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280010.png" />-dimensional smooth knots is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280011.png" />. The operation of connected sum defines on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280012.png" /> an Abelian group structure. The inverse of the knot cobordism class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280013.png" /> is the knot cobordism class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280014.png" />.
+
An equivalence relation on the set of knots, weaker than the isotopy type relation. Two smooth $  n $-
 +
dimensional knots $  K _ {1} = ( S ^ {n + 2 } , k _ {1}  ^ {n} ) $
 +
and $  K _ {2} = ( S ^ {n + 2 } , k _ {2}  ^ {n} ) $
 +
are said to be cobordant if there exists a smooth oriented $  ( n + 1 ) $-
 +
dimensional submanifold $  V $
 +
of $  [ 0, 1] \times S ^ {n + 2 } $,
 +
where $  V $
 +
is homeomorphic to $  [ 0, 1] \times S  ^ {n} $
 +
and $  \partial  V = V \cap \{ 0, 1 \} \times ( S ^ {n + 2 } ) = ( 0 \times k _ {1} ) \cup ( 1 \times - k _ {2} ) $.  
 +
Here the minus sign indicates the opposite orientation. Knots cobordant to the trivial knot are called cobordant to zero, or slice knots. The set of (cobordance) equivalence classes of $  n $-
 +
dimensional smooth knots is denoted by $  C _ {n} $.  
 +
The operation of connected sum defines on $  C _ {n} $
 +
an Abelian group structure. The inverse of the knot cobordism class $  ( S ^ {n + 2 } , k  ^ {n} ) $
 +
is the knot cobordism class $  (- S ^ {n + 2 } , - k  ^ {n} ) $.
  
For every even <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280015.png" /> the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280016.png" /> is zero. The knot cobordism class of an odd-dimensional knot is defined by its [[Seifert matrix|Seifert matrix]]. A square integral matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280017.png" /> is called cobordant to zero if it is unimodularly congruent to a matrix of the form
+
For every even $  n $
 +
the group $  C _ {n} $
 +
is zero. The knot cobordism class of an odd-dimensional knot is defined by its [[Seifert matrix|Seifert matrix]]. A square integral matrix $  A $
 +
is called cobordant to zero if it is unimodularly congruent to a matrix of the form
  
<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/c/c022/c022800/c02280018.png" /></td> </tr></table>
+
$$
 +
\left \|
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280019.png" /> are square matrices of the same dimensions and 0 is the zero matrix. Two square matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280021.png" /> are called cobordant if the matrix
+
\begin{array}{ll}
 +
0 &N _ {1}  \\
 +
N _ {2}  &N _ {3}  \\
 +
\end{array}
 +
\right \| ,
 +
$$
  
<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/c/c022/c022800/c02280022.png" /></td> </tr></table>
+
where  $  N _ {1} , N _ {2} , N _ {3} $
 +
are square matrices of the same dimensions and 0 is the zero matrix. Two square matrices  $  A _ {1} $
 +
and  $  A _ {2} $
 +
are called cobordant if the matrix
  
is cobordant to zero. A square integral matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280023.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280025.png" />-matrix, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280026.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280027.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280028.png" />. The Seifert matrix of every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280029.png" />-dimensional knot is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280030.png" />-matrix. For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280031.png" /> the cobordance relation is an equivalence relation on the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280032.png" />-matrices. The set of equivalence classes is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280033.png" />. The operation of direct sum defines on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280034.png" /> an Abelian group structure. One has the Levine homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280035.png" /> which associates with the cobordism class of the knot <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280036.png" /> the cobordism class of the Seifert matrix of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280037.png" />. The Levine homomorphism is an isomorphism for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280038.png" />. The homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280039.png" /> is a monomorphism, its image is a subgroup of index 2 in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280040.png" />, consisting of the class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280041.png" />-matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280042.png" /> for which the signature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280043.png" /> is divisible by 16. The homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280044.png" /> is an epimorphism; its kernel is non-trivial.
+
$$
 +
\left \|
 +
\begin{array}{lr}
 +
A _ {1}  & 0 \\
 +
0 &- A _ {2}  \\
 +
\end{array}
 +
\
 +
\right \|
 +
$$
  
For a study of the structure of the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280046.png" /> and for the construction of a complete system of invariants of knot cobordism classes one makes use of the following construction. An isometric structure over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280047.png" /> is a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280048.png" /> consisting of a non-degenerate quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280049.png" /> given on a finite-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280050.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280051.png" /> and an isometry <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280052.png" />. An isometric structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280053.png" /> is called cobordant to zero if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280054.png" /> contains a totally-isotropic subspace of half its dimension that is invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280055.png" />. The operation of orthogonal sum of forms and direct sum of isometries defines an operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280056.png" /> on the set of isometric structures. Two isometric structures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280058.png" /> are called cobordant if the isometric structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280059.png" /> is cobordant to zero. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280060.png" /> be the set of cobordism classes of isometric structures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280061.png" /> satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280062.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280063.png" /> is the characteristic polynomial of the isometry <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280064.png" />. In the study of the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280066.png" /> an important role is played by the imbeddings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280067.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280068.png" />, which are constructed as follows. Every cobordism class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280069.png" />-matrices contains a non-degenerate matrix. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280070.png" /> is a non-degenerate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280071.png" />-matrix, put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280073.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280074.png" /> be the isometric structure whose form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280075.png" /> has the given matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280076.png" />, while its isometry <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280077.png" /> has the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280078.png" />. This gives a well-defined homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280079.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280080.png" />.
+
is cobordant to zero. A square integral matrix  $  A $
 +
is called an  $  \epsilon $-
 +
matrix, where  $  \epsilon = + 1 $
 +
or  $  - 1 $,
 +
if  $  \mathop{\rm det} ( A + \epsilon A  ^  \prime  ) = \pm  1 $.  
 +
The Seifert matrix of every  $  ( 2q - 1) $-
 +
dimensional knot is a $  (- 1)  ^ {q} $-
 +
matrix. For every  $  \epsilon $
 +
the cobordance relation is an equivalence relation on the set of all  $  \epsilon $-
 +
matrices. The set of equivalence classes is denoted by  $  G _  \epsilon  $.  
 +
The operation of direct sum defines on  $  G _  \epsilon  $
 +
an Abelian group structure. One has the Levine homomorphism  $  \phi _ {q} : C _ {2q - 1 }  \rightarrow G _ {(- 1)  ^ {q}  } $
 +
which associates with the cobordism class of the knot  $  K $
 +
the cobordism class of the Seifert matrix of $  K $.  
 +
The Levine homomorphism is an isomorphism for all  $  q \geq  3 $.  
 +
The homomorphism  $  \phi _ {2} : C _ {3} \rightarrow G _ {+} 1 $
 +
is a monomorphism, its image is a subgroup of index 2 in  $  G _ {+} 1 $,
 +
consisting of the class of $  (+ 1) $-
 +
matrices $  A $
 +
for which the signature of  $  A + A  ^  \prime  $
 +
is divisible by 16. The homomorphism  $  \phi _ {1} : C _ {+} 1 \rightarrow G _ {-} 1 $
 +
is an epimorphism; its kernel is non-trivial.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280081.png" /> be an isometric structure on a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280082.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280083.png" />. Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280084.png" /> the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280086.png" />-primary component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280087.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280088.png" /> for large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280089.png" />. A polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280090.png" /> is called reciprocal if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280091.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280092.png" />. For each irreducible reciprocal polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280093.png" /> denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280094.png" /> the exponent, reduced modulo 2, with which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280095.png" /> divides the characteristic polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280096.png" /> of the isometry <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280097.png" />. For every reciprocal polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280098.png" /> irreducible over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280099.png" />, denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800100.png" /> the signature of the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800101.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800102.png" />. For each prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800103.png" /> and each reciprocal polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800104.png" /> irreducible over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800105.png" />, denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800106.png" /> the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800107.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800108.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800109.png" /> is the field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800110.png" />-adic numbers. Put
+
For a study of the structure of the groups  $  G _ {+} 1 $
 +
and  $  G _ {-} 1 $
 +
and for the construction of a complete system of invariants of knot cobordism classes one makes use of the following construction. An isometric structure over a field  $  F $
 +
is a pair  $  (\langle  , \rangle;  T) $
 +
consisting of a non-degenerate quadratic form  $  \langle  , \rangle $
 +
given on a finite-dimensional vector space $  V $
 +
over  $  F $
 +
and an isometry  $  T: V \rightarrow V $.  
 +
An isometric structure  $  (\langle  , \rangle;  T) $
 +
is called cobordant to zero if  $  V $
 +
contains a totally-isotropic subspace of half its dimension that is invariant under  $  T $.  
 +
The operation of orthogonal sum of forms and direct sum of isometries defines an operation  $  \perp  $
 +
on the set of isometric structures. Two isometric structures  $  (\langle  , \rangle;  T) $
 +
and  $  (\langle  , \rangle  ^  \prime  ;  T ^ { \prime } ) $
 +
are called cobordant if the isometric structure  $  (\langle  , \rangle;  T)  \perp  (- \langle  , \rangle  ^  \prime  ;  T ^ { \prime } ) $
 +
is cobordant to zero. Let  $  G _ {F} $
 +
be the set of cobordism classes of isometric structures  $  (\langle  , \rangle;  T) $
 +
satisfying the condition  $  \Delta _ {T} ( 1) \times \Delta _ {T} (- 1) \neq 0 $,  
 +
where  $  \Delta _ {T} ( t) $
 +
is the characteristic polynomial of the isometry $  T $.  
 +
In the study of the groups  $  G _ {+} 1 $
 +
and  $  G _ {-} 1 $
 +
an important role is played by the imbeddings  $  \chi _ {+} : G _ {+} 1 \rightarrow G _ {Q} $
 +
and  $  \chi _ {-} : G _ {-} 1 \rightarrow G _ {Q} $,  
 +
which are constructed as follows. Every cobordism class of $  \epsilon $-
 +
matrices contains a non-degenerate matrix. If  $  A $
 +
is a non-degenerate  $  \epsilon $-
 +
matrix, put  $  B = - A  ^ {-} 1 A  ^  \prime  $,
 +
$  Q = A + A  ^  \prime $
 +
and let  $  (\langle  , \rangle;  T) $
 +
be the isometric structure whose form  $  \langle  , \rangle $
 +
has the given matrix  $  Q $,  
 +
while its isometry  $  T $
 +
has the matrix  $  B $.  
 +
This gives a well-defined homomorphism  $  \chi _  \epsilon  $
 +
with  $  \mathop{\rm ker}  \chi _  \epsilon  = 0 $.
  
<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/c/c022/c022800/c022800111.png" /></td> </tr></table>
+
Let  $  \alpha = (\langle  , \rangle; T) $
 +
be an isometric structure on a vector space  $  V $
 +
and let  $  \lambda \in F [ t] $.
 +
Denote by  $  V _  \lambda  $
 +
the  $  \lambda $-
 +
primary component of  $  V $,
 +
i.e.  $  V _  \lambda  = \mathop{\rm ker}  \lambda ( T)  ^ {N} $
 +
for large  $  N $.
 +
A polynomial  $  \lambda ( t) = t  ^ {k} + a _ {1} t ^ {k - 1 } + \dots + 1 $
 +
is called reciprocal if  $  a _ {i} = a _ {k - i }  $
 +
for all  $  i $.
 +
For each irreducible reciprocal polynomial  $  \lambda \in \mathbf Q [ t] $
 +
denote by  $  \epsilon _  \lambda  ( \alpha ) $
 +
the exponent, reduced modulo 2, with which  $  \lambda $
 +
divides the characteristic polynomial  $  \Delta _ {T} $
 +
of the isometry  $  T $.  
 +
For every reciprocal polynomial  $  \lambda \in \mathbf R [ t] $
 +
irreducible over  $  \mathbf R [ t] $,
 +
denote by  $  \sigma _  \lambda  ( \alpha ) $
 +
the signature of the restriction of  $  \langle  , \rangle $
 +
to  $  ( V \otimes \mathbf R ) _  \lambda  $.  
 +
For each prime number  $  p $
 +
and each reciprocal polynomial  $  \lambda \in \mathbf Q _ {p} [ t] $
 +
irreducible over  $  \mathbf Q _ {p} [ t] $,
 +
denote by  $  \langle  , \rangle _  \lambda  ^ {p} $
 +
the restriction of  $  \langle  , \rangle $
 +
to  $  ( V \otimes \mathbf Q _ {p} ) _  \lambda  $,
 +
where  $  \mathbf Q _ {p} $
 +
is the field of  $  p $-
 +
adic numbers. Put
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800112.png" /> is the Hilbert symbol in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800113.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800114.png" /> is the Hasse symbol and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800115.png" /> is the rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800116.png" />. Two isometric structures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800117.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800118.png" /> are cobordant if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800119.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800120.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800121.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800122.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800123.png" /> for which these invariants are defined (cf. [[#References|[3]]], [[#References|[4]]]).
+
$$
 +
\mu _  \lambda  ^ {p} ( \alpha )  = \
 +
(- 1, 1) ^ {r ( r + 3) / 2 }
 +
(  \mathop{\rm det}  \langle  , \rangle _  \lambda  ^ {p} , - 1 )  ^ {r}
 +
S (\langle  , \rangle _  \lambda  ^ {p} ),
 +
$$
  
The compositions of the Levine homomorphism, the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800124.png" /> and the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800125.png" /> associate with every odd-dimensional knot <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800126.png" /> the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800127.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800128.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800129.png" />. Two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800130.png" />-dimensional knots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800131.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800132.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800133.png" />, are cobordant if and only if
+
where  $  (  , ) $
 +
is the Hilbert symbol in  $  \mathbf Q _ {p} $,  
 +
$  S $
 +
is the Hasse symbol and $  2r $
 +
is the rank of  $  \langle  , \rangle _  \lambda  ^ {p} $.  
 +
Two isometric structures  $  \alpha $
 +
and  $  \beta $
 +
are cobordant if and only if  $  \epsilon _  \lambda  ( \alpha ) = \epsilon _  \lambda  ( \beta ) $,
 +
$  \sigma _  \lambda  ( \alpha ) = \sigma _  \lambda  ( \beta ) $
 +
and  $  \mu _  \lambda  ^ {p} ( \alpha ) = \mu _  \lambda  ^ {p} ( \beta ) $
 +
for all  $  \lambda $
 +
and $  p $
 +
for which these invariants are defined (cf. [[#References|[3]]], [[#References|[4]]]).
  
<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/c/c022/c022800/c022800134.png" /></td> </tr></table>
+
The compositions of the Levine homomorphism, the homomorphism  $  \chi $
 +
and the functions  $  \epsilon _  \lambda  , \sigma _  \lambda  , \mu _  \lambda  ^ {p} $
 +
associate with every odd-dimensional knot  $  K $
 +
the numbers  $  \epsilon _  \lambda  ( K) \in \{ 0, 1 \} $,
 +
$  \sigma _  \lambda  ( K) \in \mathbf Z $,
 +
$  \mu _  \lambda  ^ {p} ( K) \in \{ - 1, 1 \} $.  
 +
Two  $  ( 2q - 1) $-
 +
dimensional knots  $  K _ {1} $
 +
and  $  K _ {2} $,
 +
where  $  q > 1 $,
 +
are cobordant if and only if
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800135.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800136.png" /> for which these invariants are defined. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800137.png" /> is equal to the signature of the knot <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800138.png" /> (cf. [[Knots and links, quadratic forms of|Knots and links, quadratic forms of]]), where the sum is extended over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800139.png" /> of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800140.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800141.png" />, and in the sum only a finite number of terms are distinct from zero.
+
$$
 +
\epsilon _  \lambda  ( K _ {1} )  = \
 +
\epsilon ( K _ {2} ),\ \
 +
\sigma _  \lambda  ( K _ {1} ) = \
 +
\sigma _  \lambda  ( K _ {2} ),\ \
 +
\mu _  \lambda  ^ {p}
 +
( K _ {1} )  = \
 +
\mu _  \lambda  ^ {p}
 +
( K _ {2} )
 +
$$
  
Similarly one defines the group of locally flat or piecewise-linear knot cobordisms, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800142.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800143.png" />, respectively. For all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800144.png" /> one has an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800145.png" />. The natural mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800146.png" /> is an isomorphism for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800147.png" />, while for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800148.png" /> it is a monomorphism with an image of index 2. This means, in particular, that there exists a non-smooth locally flat topologically three-dimensional knot in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800149.png" /> (cf. [[#References|[5]]]).
+
for all  $  \lambda $
 +
and $  p $
 +
for which these invariants are defined. $  \sum \sigma _  \lambda  ( K) $
 +
is equal to the signature of the knot $  K $(
 +
cf. [[Knots and links, quadratic forms of|Knots and links, quadratic forms of]]), where the sum is extended over all  $  \lambda ( t) $
 +
of the form  $  t  ^ {2} - 2t  \cos  \theta + 1 $,
 +
where  $  0 < \theta < \pi $,
 +
and in the sum only a finite number of terms are distinct from zero.
  
The theory of cobordism of knots is connected with the study of singularities of not locally flat or piecewise-linear imbeddings of codimension 2. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800150.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800151.png" />-dimensional oriented manifold, imbedded as a subcomplex in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800152.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800153.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800154.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800155.png" /> is a small star-shaped neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800156.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800157.png" />, then the singularity of the imbedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800158.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800159.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800160.png" /> may be measured as follows. The boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800161.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800162.png" />-dimensional sphere, the orientation of which is defined by that of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800163.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800164.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800165.png" />-dimensional sphere the orientation of which is defined by that of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800166.png" />. This defines an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800167.png" />-dimensional knot <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800168.png" />, called the singularity of the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800169.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800170.png" />.
+
Similarly one defines the group of locally flat or piecewise-linear knot cobordisms, denoted by  $  C _ {n} ^ { \mathop{\rm TOP} } $
 +
and  $  C _ {n} ^ { \mathop{\rm PL} } $,
 +
respectively. For all  $  n $
 +
one has an isomorphism  $  C _ {n} ^ { \mathop{\rm PL} } \approx C _ {n} $.
 +
The natural mapping  $  C _ {n} \rightarrow C _ {n} ^ { \mathop{\rm TOP} } $
 +
is an isomorphism for  $  n > 3 $,
 +
while for  $  n = 3 $
 +
it is a monomorphism with an image of index 2. This means, in particular, that there exists a non-smooth locally flat topologically three-dimensional knot in  $  S  ^ {5} $(
 +
cf. [[#References|[5]]]).
 +
 
 +
The theory of cobordism of knots is connected with the study of singularities of not locally flat or piecewise-linear imbeddings of codimension 2. If $  P $
 +
is an $  ( n + 2 ) $-
 +
dimensional oriented manifold, imbedded as a subcomplex in an $  ( n + 3 ) $-
 +
dimensional manifold $  M $,  
 +
$  x \in P $,  
 +
and $  N $
 +
is a small star-shaped neighbourhood of $  x $
 +
in $  M $,  
 +
then the singularity of the imbedding of $  P $
 +
in $  M $
 +
at $  x $
 +
may be measured as follows. The boundary $  \partial  N $
 +
is an $  ( n + 2 ) $-
 +
dimensional sphere, the orientation of which is defined by that of $  M $;  
 +
$  P \cap \partial  N $
 +
is an $  n $-
 +
dimensional sphere the orientation of which is defined by that of $  P $.  
 +
This defines an $  n $-
 +
dimensional knot $  ( \partial  N, \partial  N \cap P) $,  
 +
called the singularity of the imbedding $  P \subset  M $
 +
at the point $  x $.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R.H. Fox,   J.W. Milnor,   "Singularities of 2-spheres in 4-space and cobordism of knots" ''Osaka Math. J.'' , '''3''' (1966) pp. 257–267</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M.A. Kervaire,   "Les noeuds de dimensions supérieures" ''Bull. Soc. Math. France'' , '''93''' (1965) pp. 225–271</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J. Levine,   "Knot cobordism groups in codimension 2" ''Comment. Math. Helv.'' , '''44''' (1969) pp. 229–244</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> J. Levine,   "Invariants of knot cobordism" ''Invent. Math.'' , '''8''' (1969) pp. 98–110</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> S.E. Capell,   J.L. Shaneson,   "Topological knots and knot cobordism" ''Topology'' , '''12''' (1973) pp. 33–40</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> N.W. Stoltzfus,   "Unraveling the integral knot concordance group" ''Mem. Amer. Math. Soc.'' , '''12''' (1977) pp. 192</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R.H. Fox, J.W. Milnor, "Singularities of 2-spheres in 4-space and cobordism of knots" ''Osaka Math. J.'' , '''3''' (1966) pp. 257–267 {{MR|0211392}} {{ZBL|0146.45501}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M.A. Kervaire, "Les noeuds de dimensions supérieures" ''Bull. Soc. Math. France'' , '''93''' (1965) pp. 225–271 {{MR|0189052}} {{ZBL|0141.21201}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J. Levine, "Knot cobordism groups in codimension 2" ''Comment. Math. Helv.'' , '''44''' (1969) pp. 229–244</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> J. Levine, "Invariants of knot cobordism" ''Invent. Math.'' , '''8''' (1969) pp. 98–110 {{MR|0253348}} {{ZBL|0179.52401}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> S.E. Capell, J.L. Shaneson, "Topological knots and knot cobordism" ''Topology'' , '''12''' (1973) pp. 33–40 {{MR|321099}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> N.W. Stoltzfus, "Unraveling the integral knot concordance group" ''Mem. Amer. Math. Soc.'' , '''12''' (1977) pp. 192 {{MR|0467764}} {{ZBL|0366.57005}} </TD></TR></table>
 
 
 
 
  
 
====Comments====
 
====Comments====
Line 40: Line 241:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> L.H. Kaufmann,   "On knots" , Princeton Univ. Press (1987)</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> L.H. Kaufmann, "On knots" , Princeton Univ. Press (1987)</TD></TR></table>

Latest revision as of 17:45, 4 June 2020


knot cobordism (proper bordism of knots, see Bordism)

An equivalence relation on the set of knots, weaker than the isotopy type relation. Two smooth $ n $- dimensional knots $ K _ {1} = ( S ^ {n + 2 } , k _ {1} ^ {n} ) $ and $ K _ {2} = ( S ^ {n + 2 } , k _ {2} ^ {n} ) $ are said to be cobordant if there exists a smooth oriented $ ( n + 1 ) $- dimensional submanifold $ V $ of $ [ 0, 1] \times S ^ {n + 2 } $, where $ V $ is homeomorphic to $ [ 0, 1] \times S ^ {n} $ and $ \partial V = V \cap \{ 0, 1 \} \times ( S ^ {n + 2 } ) = ( 0 \times k _ {1} ) \cup ( 1 \times - k _ {2} ) $. Here the minus sign indicates the opposite orientation. Knots cobordant to the trivial knot are called cobordant to zero, or slice knots. The set of (cobordance) equivalence classes of $ n $- dimensional smooth knots is denoted by $ C _ {n} $. The operation of connected sum defines on $ C _ {n} $ an Abelian group structure. The inverse of the knot cobordism class $ ( S ^ {n + 2 } , k ^ {n} ) $ is the knot cobordism class $ (- S ^ {n + 2 } , - k ^ {n} ) $.

For every even $ n $ the group $ C _ {n} $ is zero. The knot cobordism class of an odd-dimensional knot is defined by its Seifert matrix. A square integral matrix $ A $ is called cobordant to zero if it is unimodularly congruent to a matrix of the form

$$ \left \| \begin{array}{ll} 0 &N _ {1} \\ N _ {2} &N _ {3} \\ \end{array} \right \| , $$

where $ N _ {1} , N _ {2} , N _ {3} $ are square matrices of the same dimensions and 0 is the zero matrix. Two square matrices $ A _ {1} $ and $ A _ {2} $ are called cobordant if the matrix

$$ \left \| \begin{array}{lr} A _ {1} & 0 \\ 0 &- A _ {2} \\ \end{array} \ \right \| $$

is cobordant to zero. A square integral matrix $ A $ is called an $ \epsilon $- matrix, where $ \epsilon = + 1 $ or $ - 1 $, if $ \mathop{\rm det} ( A + \epsilon A ^ \prime ) = \pm 1 $. The Seifert matrix of every $ ( 2q - 1) $- dimensional knot is a $ (- 1) ^ {q} $- matrix. For every $ \epsilon $ the cobordance relation is an equivalence relation on the set of all $ \epsilon $- matrices. The set of equivalence classes is denoted by $ G _ \epsilon $. The operation of direct sum defines on $ G _ \epsilon $ an Abelian group structure. One has the Levine homomorphism $ \phi _ {q} : C _ {2q - 1 } \rightarrow G _ {(- 1) ^ {q} } $ which associates with the cobordism class of the knot $ K $ the cobordism class of the Seifert matrix of $ K $. The Levine homomorphism is an isomorphism for all $ q \geq 3 $. The homomorphism $ \phi _ {2} : C _ {3} \rightarrow G _ {+} 1 $ is a monomorphism, its image is a subgroup of index 2 in $ G _ {+} 1 $, consisting of the class of $ (+ 1) $- matrices $ A $ for which the signature of $ A + A ^ \prime $ is divisible by 16. The homomorphism $ \phi _ {1} : C _ {+} 1 \rightarrow G _ {-} 1 $ is an epimorphism; its kernel is non-trivial.

For a study of the structure of the groups $ G _ {+} 1 $ and $ G _ {-} 1 $ and for the construction of a complete system of invariants of knot cobordism classes one makes use of the following construction. An isometric structure over a field $ F $ is a pair $ (\langle , \rangle; T) $ consisting of a non-degenerate quadratic form $ \langle , \rangle $ given on a finite-dimensional vector space $ V $ over $ F $ and an isometry $ T: V \rightarrow V $. An isometric structure $ (\langle , \rangle; T) $ is called cobordant to zero if $ V $ contains a totally-isotropic subspace of half its dimension that is invariant under $ T $. The operation of orthogonal sum of forms and direct sum of isometries defines an operation $ \perp $ on the set of isometric structures. Two isometric structures $ (\langle , \rangle; T) $ and $ (\langle , \rangle ^ \prime ; T ^ { \prime } ) $ are called cobordant if the isometric structure $ (\langle , \rangle; T) \perp (- \langle , \rangle ^ \prime ; T ^ { \prime } ) $ is cobordant to zero. Let $ G _ {F} $ be the set of cobordism classes of isometric structures $ (\langle , \rangle; T) $ satisfying the condition $ \Delta _ {T} ( 1) \times \Delta _ {T} (- 1) \neq 0 $, where $ \Delta _ {T} ( t) $ is the characteristic polynomial of the isometry $ T $. In the study of the groups $ G _ {+} 1 $ and $ G _ {-} 1 $ an important role is played by the imbeddings $ \chi _ {+} : G _ {+} 1 \rightarrow G _ {Q} $ and $ \chi _ {-} : G _ {-} 1 \rightarrow G _ {Q} $, which are constructed as follows. Every cobordism class of $ \epsilon $- matrices contains a non-degenerate matrix. If $ A $ is a non-degenerate $ \epsilon $- matrix, put $ B = - A ^ {-} 1 A ^ \prime $, $ Q = A + A ^ \prime $ and let $ (\langle , \rangle; T) $ be the isometric structure whose form $ \langle , \rangle $ has the given matrix $ Q $, while its isometry $ T $ has the matrix $ B $. This gives a well-defined homomorphism $ \chi _ \epsilon $ with $ \mathop{\rm ker} \chi _ \epsilon = 0 $.

Let $ \alpha = (\langle , \rangle; T) $ be an isometric structure on a vector space $ V $ and let $ \lambda \in F [ t] $. Denote by $ V _ \lambda $ the $ \lambda $- primary component of $ V $, i.e. $ V _ \lambda = \mathop{\rm ker} \lambda ( T) ^ {N} $ for large $ N $. A polynomial $ \lambda ( t) = t ^ {k} + a _ {1} t ^ {k - 1 } + \dots + 1 $ is called reciprocal if $ a _ {i} = a _ {k - i } $ for all $ i $. For each irreducible reciprocal polynomial $ \lambda \in \mathbf Q [ t] $ denote by $ \epsilon _ \lambda ( \alpha ) $ the exponent, reduced modulo 2, with which $ \lambda $ divides the characteristic polynomial $ \Delta _ {T} $ of the isometry $ T $. For every reciprocal polynomial $ \lambda \in \mathbf R [ t] $ irreducible over $ \mathbf R [ t] $, denote by $ \sigma _ \lambda ( \alpha ) $ the signature of the restriction of $ \langle , \rangle $ to $ ( V \otimes \mathbf R ) _ \lambda $. For each prime number $ p $ and each reciprocal polynomial $ \lambda \in \mathbf Q _ {p} [ t] $ irreducible over $ \mathbf Q _ {p} [ t] $, denote by $ \langle , \rangle _ \lambda ^ {p} $ the restriction of $ \langle , \rangle $ to $ ( V \otimes \mathbf Q _ {p} ) _ \lambda $, where $ \mathbf Q _ {p} $ is the field of $ p $- adic numbers. Put

$$ \mu _ \lambda ^ {p} ( \alpha ) = \ (- 1, 1) ^ {r ( r + 3) / 2 } ( \mathop{\rm det} \langle , \rangle _ \lambda ^ {p} , - 1 ) ^ {r} S (\langle , \rangle _ \lambda ^ {p} ), $$

where $ ( , ) $ is the Hilbert symbol in $ \mathbf Q _ {p} $, $ S $ is the Hasse symbol and $ 2r $ is the rank of $ \langle , \rangle _ \lambda ^ {p} $. Two isometric structures $ \alpha $ and $ \beta $ are cobordant if and only if $ \epsilon _ \lambda ( \alpha ) = \epsilon _ \lambda ( \beta ) $, $ \sigma _ \lambda ( \alpha ) = \sigma _ \lambda ( \beta ) $ and $ \mu _ \lambda ^ {p} ( \alpha ) = \mu _ \lambda ^ {p} ( \beta ) $ for all $ \lambda $ and $ p $ for which these invariants are defined (cf. [3], [4]).

The compositions of the Levine homomorphism, the homomorphism $ \chi $ and the functions $ \epsilon _ \lambda , \sigma _ \lambda , \mu _ \lambda ^ {p} $ associate with every odd-dimensional knot $ K $ the numbers $ \epsilon _ \lambda ( K) \in \{ 0, 1 \} $, $ \sigma _ \lambda ( K) \in \mathbf Z $, $ \mu _ \lambda ^ {p} ( K) \in \{ - 1, 1 \} $. Two $ ( 2q - 1) $- dimensional knots $ K _ {1} $ and $ K _ {2} $, where $ q > 1 $, are cobordant if and only if

$$ \epsilon _ \lambda ( K _ {1} ) = \ \epsilon ( K _ {2} ),\ \ \sigma _ \lambda ( K _ {1} ) = \ \sigma _ \lambda ( K _ {2} ),\ \ \mu _ \lambda ^ {p} ( K _ {1} ) = \ \mu _ \lambda ^ {p} ( K _ {2} ) $$

for all $ \lambda $ and $ p $ for which these invariants are defined. $ \sum \sigma _ \lambda ( K) $ is equal to the signature of the knot $ K $( cf. Knots and links, quadratic forms of), where the sum is extended over all $ \lambda ( t) $ of the form $ t ^ {2} - 2t \cos \theta + 1 $, where $ 0 < \theta < \pi $, and in the sum only a finite number of terms are distinct from zero.

Similarly one defines the group of locally flat or piecewise-linear knot cobordisms, denoted by $ C _ {n} ^ { \mathop{\rm TOP} } $ and $ C _ {n} ^ { \mathop{\rm PL} } $, respectively. For all $ n $ one has an isomorphism $ C _ {n} ^ { \mathop{\rm PL} } \approx C _ {n} $. The natural mapping $ C _ {n} \rightarrow C _ {n} ^ { \mathop{\rm TOP} } $ is an isomorphism for $ n > 3 $, while for $ n = 3 $ it is a monomorphism with an image of index 2. This means, in particular, that there exists a non-smooth locally flat topologically three-dimensional knot in $ S ^ {5} $( cf. [5]).

The theory of cobordism of knots is connected with the study of singularities of not locally flat or piecewise-linear imbeddings of codimension 2. If $ P $ is an $ ( n + 2 ) $- dimensional oriented manifold, imbedded as a subcomplex in an $ ( n + 3 ) $- dimensional manifold $ M $, $ x \in P $, and $ N $ is a small star-shaped neighbourhood of $ x $ in $ M $, then the singularity of the imbedding of $ P $ in $ M $ at $ x $ may be measured as follows. The boundary $ \partial N $ is an $ ( n + 2 ) $- dimensional sphere, the orientation of which is defined by that of $ M $; $ P \cap \partial N $ is an $ n $- dimensional sphere the orientation of which is defined by that of $ P $. This defines an $ n $- dimensional knot $ ( \partial N, \partial N \cap P) $, called the singularity of the imbedding $ P \subset M $ at the point $ x $.

References

[1] R.H. Fox, J.W. Milnor, "Singularities of 2-spheres in 4-space and cobordism of knots" Osaka Math. J. , 3 (1966) pp. 257–267 MR0211392 Zbl 0146.45501
[2] M.A. Kervaire, "Les noeuds de dimensions supérieures" Bull. Soc. Math. France , 93 (1965) pp. 225–271 MR0189052 Zbl 0141.21201
[3] J. Levine, "Knot cobordism groups in codimension 2" Comment. Math. Helv. , 44 (1969) pp. 229–244
[4] J. Levine, "Invariants of knot cobordism" Invent. Math. , 8 (1969) pp. 98–110 MR0253348 Zbl 0179.52401
[5] S.E. Capell, J.L. Shaneson, "Topological knots and knot cobordism" Topology , 12 (1973) pp. 33–40 MR321099
[6] N.W. Stoltzfus, "Unraveling the integral knot concordance group" Mem. Amer. Math. Soc. , 12 (1977) pp. 192 MR0467764 Zbl 0366.57005

Comments

Another term for cobordance of knots is concordance of knots, and correspondingly one has the knot concordance group.

References

[a1] L.H. Kaufmann, "On knots" , Princeton Univ. Press (1987)
How to Cite This Entry:
Cobordism of knots. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cobordism_of_knots&oldid=14400
This article was adapted from an original article by M.Sh. Farber (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article