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An isotopy class of imbeddings of a sphere into a sphere. More precisely, an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m0651303.png" />-dimensional knot of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m0651304.png" /> is a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m0651305.png" /> consisting of an oriented sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m0651306.png" /> and an oriented, locally flat, submanifold of it, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m0651307.png" />, homeomorphic to the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m0651308.png" />. Two knots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m0651309.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513010.png" /> are called equivalent if there is an [[Isotopy (in topology)|isotopy (in topology)]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513011.png" /> which takes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513012.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513013.png" /> while preserving the orientation. Depending on the category (Diff, PL or Top) from which the terms "submanifold" and "isotopy" in these definitions are taken, one speaks of smooth, piecewise-linear or topological multi-dimensional knots, respectively. In the smooth case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513014.png" /> may have a non-standard differentiable structure. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513015.png" />-dimensional knot of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513016.png" /> which is isotopic to the standard imbedding is called a trivial, or unknotted, knot.
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The study of multi-dimensional knots of codimension 1 is related to the [[Schoenflies conjecture|Schoenflies conjecture]]. Every topological knot of codimension 1 is trivial. This is true for piecewise-linear and smooth knots if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513017.png" />.
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Piecewise-linear and topological multi-dimensional knots of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513018.png" /> are trivial. In the smooth case this is not so. The set of isotopy classes of smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513019.png" />-dimensional knots of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513020.png" /> coincides, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513021.png" />, with the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513022.png" /> of cobordism classes of knots. (Two multi-dimensional knots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513024.png" /> are called cobordant if there is a smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513025.png" />-dimensional submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513026.png" /> transversal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513027.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513029.png" /> is an [[H-cobordism|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513030.png" />-cobordism]] between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513032.png" />.) The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513033.png" /> is an Abelian group with respect to the operation of connected sum. In this group the negative of the class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513034.png" /> is the cobordism class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513035.png" />, where the minus denotes reversal of orientation. There is a natural homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513036.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513037.png" /> is the group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513038.png" />-dimensional homotopy spheres; this homomorphism associates the differentiable structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513039.png" /> to the knot <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513040.png" />. The kernel of this homomorphism, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513041.png" />, is the set of isotopy classes of the standard sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513042.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513043.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513044.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513045.png" /> is trivial. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513047.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513048.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513050.png" /> are finite. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513052.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513053.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513055.png" /> are finitely-generated Abelian groups of rank 1 (see [[#References|[1]]], [[#References|[2]]]). The set of concordance classes of smooth imbeddings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513056.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513057.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513058.png" /> has also been calculated (see [[#References|[3]]]).
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An isotopy class of imbeddings of a sphere into a sphere. More precisely, an  $  n $-
 +
dimensional knot of codimension $  q $
 +
is a pair  $  K = ( S  ^ {n+} q , k  ^ {n} ) $
 +
consisting of an oriented sphere  $  S  ^ {n+} q $
 +
and an oriented, locally flat, submanifold of it, $  k  ^ {n} $,  
 +
homeomorphic to the sphere  $  S  ^ {n} $.  
 +
Two knots $  K _ {1} ( S  ^ {n+} q , k _ {1}  ^ {n} ) $
 +
and $  K _ {2} = ( S  ^ {n+} q , k _ {2}  ^ {n} ) $
 +
are called equivalent if there is an [[Isotopy (in topology)|isotopy (in topology)]] of  $  S  ^ {n+} q $
 +
which takes  $  k _ {1}  ^ {n} $
 +
to $  k _ {2}  ^ {n} $
 +
while preserving the orientation. Depending on the category (Diff, PL or Top) from which the terms "submanifold" and "isotopy" in these definitions are taken, one speaks of smooth, piecewise-linear or topological multi-dimensional knots, respectively. In the smooth case  $  k  ^ {n} $
 +
may have a non-standard differentiable structure. An  $  n $-
 +
dimensional knot of codimension  $  q $
 +
which is isotopic to the standard imbedding is called a trivial, or unknotted, knot.
  
The study of multi-dimensional knots of codimension 2, which will subsequently simply be called knots, proceeds quite similarly in all three categories (Diff, PL, Top). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513059.png" /> every topological knot may be transformed by an isotopy to a smooth knot. However, there are topological three-dimensional knots in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513060.png" /> which are not equivalent, or even cobordant, to smooth knots (see [[#References|[4]]]).
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The study of multi-dimensional knots of codimension 1 is related to the [[Schoenflies conjecture|Schoenflies conjecture]]. Every topological knot of codimension 1 is trivial. This is true for piecewise-linear and smooth knots if  $  n \neq 3 , 4 $.
  
The set of isotopy classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513061.png" />-dimensional knots (in each category) is an Abelian semi-group with respect to the operation of connected sum. It is known that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513062.png" /> every element in this semi-group is a finite sum of primes, and such a decomposition is unique.
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Piecewise-linear and topological multi-dimensional knots of codimension  $  q \geq  3 $
 +
are trivial. In the smooth case this is not so. The set of isotopy classes of smooth  $  n $-
 +
dimensional knots of codimension  $  q \geq  3 $
 +
coincides, for  $  n \geq  5 $,
 +
with the set  $  \theta  ^ {n+} q,n $
 +
of cobordism classes of knots. (Two multi-dimensional knots  $  K _ {1} = ( S  ^ {n+} q , k _ {1}  ^ {n} ) $
 +
and  $  K _ {2} = ( S  ^ {n+} q , k _ {2}  ^ {n} ) $
 +
are called cobordant if there is a smooth  $  ( n + 1 ) $-
 +
dimensional submanifold  $  W \subset  S  ^ {n+} q \times I $
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transversal to  $  \partial  ( S  ^ {n+} q \times I ) $,
 +
where  $  \partial  W = ( k _ {1}  ^ {n} \times 0 ) \cup ( - k _ {2}  ^ {n} \times 1 ) $
 +
and  $  W $
 +
is an [[H-cobordism| $  h $-
 +
cobordism]] between  $  k _ {1}  ^ {n} \times 0 $
 +
and  $  k _ {2}  ^ {n} \times 1 $.)  
 +
The set  $  \theta  ^ {n+} q,n $
 +
is an Abelian group with respect to the operation of connected sum. In this group the negative of the class of  $  ( S  ^ {n+} q , k  ^ {n} ) $
 +
is the cobordism class of  $  ( - S  ^ {n+} q , - k  ^ {n} ) $,
 +
where the minus denotes reversal of orientation. There is a natural homomorphism  $  \theta  ^ {n+} q,n \rightarrow \theta  ^ {n} $,
 +
where  $  \theta  ^ {n} $
 +
is the group of  $  n $-
 +
dimensional homotopy spheres; this homomorphism associates the differentiable structure of  $  k  ^ {n} $
 +
to the knot  $  ( S  ^ {n+} q , k  ^ {n} ) $.  
 +
The kernel of this homomorphism, denoted by  $  \Sigma  ^ {n+} q,n $,
 +
is the set of isotopy classes of the standard sphere  $  S  ^ {n} $
 +
in  $  S  ^ {n+} q $.  
 +
If  $  2 q > n + 3 $,
 +
then  $  \Sigma  ^ {n+} q,n $
 +
is trivial. If  $  2 q \geq  n + 3 $
 +
and  $  ( n + 1 ) \not\equiv 0 $(
 +
$  \mathop{\rm mod}  4 $),
 +
then  $  \theta  ^ {n+} q,n $
 +
and  $  \Sigma  ^ {n+} q,n $
 +
are finite. When  $  2 q \leq  n + 3 $
 +
and  $  ( n + 1 ) \not\equiv 0 $(
 +
$  \mathop{\rm mod}  4 $),
 +
then  $  \theta  ^ {n+} q,n $
 +
and  $  \Sigma  ^ {n+} q,n $
 +
are finitely-generated Abelian groups of rank 1 (see [[#References|[1]]], [[#References|[2]]]). The set of concordance classes of smooth imbeddings of  $  S  ^ {n} $
 +
into  $  S  ^ {n+} q $
 +
for  $  q > 2 $
 +
has also been calculated (see [[#References|[3]]]).
  
An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513063.png" />-dimensional knot <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513064.png" /> is trivial if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513065.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513066.png" />. An algebraic classification has been given (see [[#References|[6]]]) of the knots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513067.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513068.png" />, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513069.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513070.png" /> odd (knots of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513072.png" />): For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513073.png" /> the set of isotopy classes of such knots turns out to be in one-to-one correspondence with the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513074.png" />-equivalence classes of the [[Seifert matrix|Seifert matrix]]. Knots of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513075.png" /> are important from the point of view of applications to algebraic geometry, since they contain all knots obtained by the following construction (see [[#References|[15]]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513076.png" /> be a complex polynomial of non-zero degree having zero as an isolated singularity and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513077.png" />. The intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513078.png" /> of the hyperplane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513079.png" /> with a small sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513080.png" /> with centre at zero is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513081.png" />-connected <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513082.png" />-dimensional manifold. The manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513083.png" /> is homeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513084.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513085.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513086.png" /> is the Alexander polynomial. In this case there thus arises a knot <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513087.png" />. Such knots are called algebraic; they are all of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513088.png" />.
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The study of multi-dimensional knots of codimension 2, which will subsequently simply be called knots, proceeds quite similarly in all three categories (Diff, PL, Top). For $  n \geq  5 $
 +
every topological knot may be transformed by an isotopy to a smooth knot. However, there are topological three-dimensional knots in  $  S  ^ {5} $
 +
which are not equivalent, or even cobordant, to smooth knots (see [[#References|[4]]]).
  
The exterior of a smooth knot <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513089.png" /> is the complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513090.png" /> (of an open tubular neighbourhood) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513091.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513092.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513093.png" />, for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513094.png" />-dimensional knot <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513095.png" /> there is a knot <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513096.png" /> such that each knot with exterior diffeomorphic to the exterior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513097.png" /> is equivalent to either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513098.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m06513099.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130100.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130101.png" /> are the exteriors of two smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130102.png" />-dimensional knots, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130103.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130104.png" />, then the following statements are equivalent (see [[#References|[7]]]): 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130106.png" /> are diffeomorphic; and 2) the pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130107.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130108.png" /> are homotopically equivalent. These results reduce the classification problem for knots to the homotopy classification of pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130109.png" /> and the solution of the question: Does the exterior determine the type of a knot, that is, does <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130110.png" /> hold? It is known that this equality holds for knots of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130111.png" /> (see [[#References|[6]]]) and for knots obtained by the Artin construction and the supertwisting construction (see [[#References|[8]]]). However, two-dimensional knots have been found in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130112.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130113.png" /> (see [[#References|[9]]]).
+
The set of isotopy classes of $  n $-
 +
dimensional knots (in each category) is an Abelian semi-group with respect to the operation of connected sum. It is known that for  $  n = 1 $
 +
every element in this semi-group is a finite sum of primes, and such a decomposition is unique.
  
The study of the homotopy type of the exterior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130114.png" /> is complicated because this exterior is not simply connected. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130115.png" /> is the group of the knot (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130116.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130117.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130118.png" />, and the weight of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130119.png" /> (that is, the minimal number of elements not contained in a proper normal divisor) is equal to 1. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130120.png" /> these properties completely describe the class of groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130121.png" />-dimensional knots (see [[#References|[10]]]). The groups of one-dimensional and two-dimensional knots have a number of additional properties (see [[Knot theory|Knot theory]]; [[Two-dimensional knot|Two-dimensional knot]]).
+
An  $  n $-
 +
dimensional knot  $  K = ( S  ^ {n+} 2 , k  ^ {n} ) $
 +
is trivial if and only if  $  \pi _ {i} ( S  ^ {n+} 2 \setminus  k  ^ {n} ) = \pi _ {i} ( S  ^ {1} ) $
 +
for all  $  i \leq  [ ( n + 1 ) / 2 ] $.  
 +
An algebraic classification has been given (see [[#References|[6]]]) of the knots  $  K $
 +
for which  $  \pi _ {i} ( S  ^ {n+} 2 \setminus  k  ^ {n} ) = \pi _ {i} ( S  ^ {1} ) $,  
 +
for all  $  i \leq  [( n+ 1)/2]- 1 $
 +
and $  n $
 +
odd (knots of type  $  L $):  
 +
For  $  n \geq  5 $
 +
the set of isotopy classes of such knots turns out to be in one-to-one correspondence with the set of  $  S $-
 +
equivalence classes of the [[Seifert matrix|Seifert matrix]]. Knots of type  $  L $
 +
are important from the point of view of applications to algebraic geometry, since they contain all knots obtained by the following construction (see [[#References|[15]]]). Let  $  f ( z _ {1} \dots z _ {q+} 1 ) $
 +
be a complex polynomial of non-zero degree having zero as an isolated singularity and let  $  f ( 0) = 0 $.
 +
The intersection  $  k $
 +
of the hyperplane  $  V = f ^ { - 1 } ( 0) $
 +
with a small sphere  $  S  ^ {q+} 1 $
 +
with centre at zero is a $  ( q - 2 ) $-
 +
connected  $  ( 2 q - 1 ) $-
 +
dimensional manifold. The manifold  $  k $
 +
is homeomorphic to  $  S  ^ {2q-} 1 $
 +
if and only if  $  | \Delta ( 1) | = 1 $,
 +
where  $  \Delta ( t) $
 +
is the Alexander polynomial. In this case there thus arises a knot $  ( S  ^ {2q+} 1 , k ) $.
 +
Such knots are called algebraic; they are all of type  $  L $.
  
Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130122.png" />, the exterior <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130123.png" /> has a unique infinite cyclic covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130124.png" />. The homology spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130125.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130126.png" />-modules. Their [[Alexander invariants|Alexander invariants]] are invariants of the knot. For algebraic properties of the modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130127.png" /> see [[#References|[10]]][[#References|[13]]].
+
The exterior of a smooth knot  $  K = ( S  ^ {n+} 2 , k  ^ {n} ) $
 +
is the complement  $  X $(
 +
of an open tubular neighbourhood) of  $  k  ^ {n} $
 +
in  $  S  ^ {n+} 2 $.  
 +
For  $  n \geq  2 $,  
 +
for each  $  n $-
 +
dimensional knot  $  K $
 +
there is a knot  $  \tau ( K) $
 +
such that each knot with exterior diffeomorphic to the exterior of  $  K $
 +
is equivalent to either  $  K $
 +
or  $  \tau ( K) $.  
 +
If  $  X _ {1} $,
 +
$  X _ {2} $
 +
are the exteriors of two smooth  $  n $-
 +
dimensional knots,  $  n \geq  3 $,
 +
and  $  \pi _ {1} ( X _ {1} ) = \pi _ {1} ( X _ {2} ) = \mathbf Z $,
 +
then the following statements are equivalent (see [[#References|[7]]]): 1)  $  X _ {1} $
 +
and  $  X _ {2} $
 +
are diffeomorphic; and 2) the pairs  $  ( X _ {1} , \partial  X _ {1} ) $
 +
and  $  ( X _ {2} , \partial  X _ {2} ) $
 +
are homotopically equivalent. These results reduce the classification problem for knots to the homotopy classification of pairs  $  ( X , \partial  X ) $
 +
and the solution of the question: Does the exterior determine the type of a knot, that is, does  $  K = \tau ( K) $
 +
hold? It is known that this equality holds for knots of type  $  L $(
 +
see [[#References|[6]]]) and for knots obtained by the Artin construction and the supertwisting construction (see [[#References|[8]]]). However, two-dimensional knots have been found in  $  S  ^ {4} $
 +
for which  $  K \neq \tau ( K) $(
 +
see [[#References|[9]]]).
  
Due to the fact that the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130128.png" /> acts without fixed points on an infinite cyclic covering, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130129.png" />-dimensional non-compact manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130130.png" /> has a number of the homological properties of compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130131.png" />-dimensional manifolds. In particular, for the homology of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130132.png" /> with coefficients from a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130133.png" /> there is a non-degenerate pairing
+
The study of the homotopy type of the exterior of  $  X $
 +
is complicated because this exterior is not simply connected. If  $  G $
 +
is the group of the knot (that is,  $  G = \pi _ {1} ( X) $),
 +
then  $  G / [ G , G ] = \mathbf Z $,
 +
$  H _ {2} ( G) = 0 $,
 +
and the weight of  $  G $(
 +
that is, the minimal number of elements not contained in a proper normal divisor) is equal to 1. For  $  n \geq  3 $
 +
these properties completely describe the class of groups of  $  n $-
 +
dimensional knots (see [[#References|[10]]]). The groups of one-dimensional and two-dimensional knots have a number of additional properties (see [[Knot theory|Knot theory]]; [[Two-dimensional knot|Two-dimensional knot]]).
  
<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/m/m065/m065130/m065130134.png" /></td> </tr></table>
+
Since  $  H  ^ {1} ( X ; \mathbf Z ) = \mathbf Z $,
 +
the exterior  $  X $
 +
has a unique infinite cyclic covering  $  p : \widetilde{X}  \rightarrow X $.
 +
The homology spaces  $  H _ {*} ( \widetilde{X}  ; \mathbf Z ) $
 +
are  $  \mathbf Z [ \mathbf Z ] $-
 +
modules. Their [[Alexander invariants|Alexander invariants]] are invariants of the knot. For algebraic properties of the modules  $  H _ {*} ( \widetilde{X}  ;  \mathbf Z ) $
 +
see [[#References|[10]]]–[[#References|[13]]].
  
with properties resembling the pairing determined by the [[Intersection index (in homology)|intersection index (in homology)]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130135.png" />-dimensional compact manifolds. There is also a pairing
+
Due to the fact that the group  $  \mathbf Z $
 +
acts without fixed points on an infinite cyclic covering, the ( n + 2 ) $-
 +
dimensional non-compact manifold  $  \widetilde{X}  $
 +
has a number of the homological properties of compact  $  ( n + 1 ) $-
 +
dimensional manifolds. In particular, for the homology of the manifold  $  \widetilde{X}  $
 +
with coefficients from a field  $  F $
 +
there is a non-degenerate 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/m/m065/m065130/m065130136.png" /></td> </tr></table>
+
$$
 +
H _ {n} ( \widetilde{X}  ; F  ) \otimes H _ {n+} 1- k
 +
( \widetilde{X}  ; F  )  \rightarrow  F ,\  k = 1 \dots n ,
 +
$$
  
similar to the linking coefficients (cf. [[Linking coefficient|Linking coefficient]]) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130137.png" />-dimensional manifolds (see [[#References|[13]]]), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130138.png" />. These homology pairings generate invariants of the homotopy type of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130139.png" />. To obtain algebraic invariants, finite-sheeted cyclic branched coverings are also used (see [[#References|[14]]]).
+
with properties resembling the pairing determined by the [[Intersection index (in homology)|intersection index (in homology)]] in  $  ( n + 1 ) $-
 +
dimensional compact manifolds. There is also a pairing
  
The problem of classifying knots of codimension 2 up to cobordism, a coarser equivalence relation than isotopy type, has been completely solved for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130140.png" /> (see [[Cobordism of knots|Cobordism of knots]]).
+
$$
 +
T _ {k} \widetilde{X}  \otimes T _ {n-} k \widetilde{X}  \rightarrow  \mathbf Q / \mathbf Z ,\ \
 +
k = 1 \dots n - 1 ,
 +
$$
 +
 
 +
similar to the linking coefficients (cf. [[Linking coefficient|Linking coefficient]]) in  $  ( n + 1 ) $-
 +
dimensional manifolds (see [[#References|[13]]]), where  $  T _ {j} \widetilde{X}  =  \mathop{\rm Tors}  H _ {j} ( \widetilde{X}  ;  \mathbf Z ) $.
 +
These homology pairings generate invariants of the homotopy type of the pair  $  ( X , \partial  X) $.
 +
To obtain algebraic invariants, finite-sheeted cyclic branched coverings are also used (see [[#References|[14]]]).
 +
 
 +
The problem of classifying knots of codimension 2 up to cobordism, a coarser equivalence relation than isotopy type, has been completely solved for $  n > 1 $(
 +
see [[Cobordism of knots|Cobordism of knots]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Haefliger, "Knotted (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130141.png" />)-spheres in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130142.png" />-space" ''Ann. of Math.'' , '''75''' (1962) pp. 452–466 {{MR|145539}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Haefliger, "Differentiable embeddings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130143.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130144.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130145.png" />" ''Ann. of Math.'' , '''83''' (1966) pp. 402–436 {{MR|}} {{ZBL|0151.32502}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J. Levine, "A classification of differentiable knots" ''Ann. of Math.'' , '''82''' (1965) pp. 15–50 {{MR|0180981}} {{ZBL|0136.21102}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> S. Cappell, J. Shaneson, "Topological knots and knot cobordism" ''Topology'' , '''12''' (1973) pp. 33–40 {{MR|0321099}} {{ZBL|0268.57006}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> A.B. Sossinskii, "Decomposition of knots" ''Math. USSR Sb.'' , '''10''' (1970) pp. 139–150 ''Mat. Sb.'' , '''81''' : 1 (1970) pp. 145–158</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> J. Levine, "An algebraic classification of some knots of codimension two" ''Comment. Math. Helv.'' , '''45''' (1970) pp. 185–198 {{MR|0266226}} {{ZBL|0211.55902}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> R. Lashof, J. Shaneson, "Classification of knots in codimension two" ''Bull. Amer. Math. Soc.'' , '''75''' (1969) pp. 171–175 {{MR|0242175}} {{ZBL|0198.28701}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> S. Cappell, "Superspinning and knot complements" J.C. Cantrell (ed.) C.H. Edwards jr. (ed.) , ''Topology of manifolds'' , Markham (1971) pp. 358–383 {{MR|0276972}} {{ZBL|0281.57001}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> S. Cappell, J. Shaneson, "There exist inequivalent knots with the same complements" ''Ann. of Math.'' , '''103''' (1976) pp. 349–353 {{MR|0413117}} {{ZBL|}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> M. 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">[11]</TD> <TD valign="top"> J. Levine, "Polynomial invariants of knots of codimension two" ''Ann. of Math.'' , '''84''' (1966) pp. 537–554 {{MR|0200922}} {{ZBL|0196.55905}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> J. Levine, "Knot modules" , ''Knots, Groups and 3-Manifolds'' , Princeton Univ. Press (1975) pp. 25–34 {{MR|0405437}} {{ZBL|0336.57008}} </TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> M.Sh. Farber, "Duality in an infinite cyclic covering and even-dimensional knots" ''Math. USSR Izv.'' , '''11''' (1974) pp. 749–781 ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''41''' (1977) pp. 794–828 {{MR|0515677}} {{ZBL|0394.57011}} </TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> O.Ya. Viro, "Branched coverings of manifolds with boundary and link invariants I" ''Math. USSR Izv.'' , '''7''' (1973) pp. 1239–1256 ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''37''' (1973) pp. 1242–1258 {{MR|}} {{ZBL|0295.55002}} </TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top"> J. Milnor, "Singular points of complex hypersurfaces" , Princeton Univ. Press (1968) {{MR|0239612}} {{ZBL|0184.48405}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Haefliger, "Knotted (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130141.png" />)-spheres in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130142.png" />-space" ''Ann. of Math.'' , '''75''' (1962) pp. 452–466 {{MR|145539}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Haefliger, "Differentiable embeddings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130143.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130144.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065130/m065130145.png" />" ''Ann. of Math.'' , '''83''' (1966) pp. 402–436 {{MR|}} {{ZBL|0151.32502}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J. Levine, "A classification of differentiable knots" ''Ann. of Math.'' , '''82''' (1965) pp. 15–50 {{MR|0180981}} {{ZBL|0136.21102}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> S. Cappell, J. Shaneson, "Topological knots and knot cobordism" ''Topology'' , '''12''' (1973) pp. 33–40 {{MR|0321099}} {{ZBL|0268.57006}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> A.B. Sossinskii, "Decomposition of knots" ''Math. USSR Sb.'' , '''10''' (1970) pp. 139–150 ''Mat. Sb.'' , '''81''' : 1 (1970) pp. 145–158</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> J. Levine, "An algebraic classification of some knots of codimension two" ''Comment. Math. Helv.'' , '''45''' (1970) pp. 185–198 {{MR|0266226}} {{ZBL|0211.55902}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> R. Lashof, J. Shaneson, "Classification of knots in codimension two" ''Bull. Amer. Math. Soc.'' , '''75''' (1969) pp. 171–175 {{MR|0242175}} {{ZBL|0198.28701}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> S. Cappell, "Superspinning and knot complements" J.C. Cantrell (ed.) C.H. Edwards jr. (ed.) , ''Topology of manifolds'' , Markham (1971) pp. 358–383 {{MR|0276972}} {{ZBL|0281.57001}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> S. Cappell, J. Shaneson, "There exist inequivalent knots with the same complements" ''Ann. of Math.'' , '''103''' (1976) pp. 349–353 {{MR|0413117}} {{ZBL|}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> M. 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">[11]</TD> <TD valign="top"> J. Levine, "Polynomial invariants of knots of codimension two" ''Ann. of Math.'' , '''84''' (1966) pp. 537–554 {{MR|0200922}} {{ZBL|0196.55905}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> J. Levine, "Knot modules" , ''Knots, Groups and 3-Manifolds'' , Princeton Univ. Press (1975) pp. 25–34 {{MR|0405437}} {{ZBL|0336.57008}} </TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> M.Sh. Farber, "Duality in an infinite cyclic covering and even-dimensional knots" ''Math. USSR Izv.'' , '''11''' (1974) pp. 749–781 ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''41''' (1977) pp. 794–828 {{MR|0515677}} {{ZBL|0394.57011}} </TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> O.Ya. Viro, "Branched coverings of manifolds with boundary and link invariants I" ''Math. USSR Izv.'' , '''7''' (1973) pp. 1239–1256 ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''37''' (1973) pp. 1242–1258 {{MR|}} {{ZBL|0295.55002}} </TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top"> J. Milnor, "Singular points of complex hypersurfaces" , Princeton Univ. Press (1968) {{MR|0239612}} {{ZBL|0184.48405}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.W. Milnor, "Infinite cyclic coverings" J. Hocking (ed.) , ''Conf. Topology of Manifolds'' , Prindle, Weber &amp; Schmidt (1968) pp. 115–133 {{MR|0242163}} {{ZBL|0179.52302}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.W. Milnor, "Infinite cyclic coverings" J. Hocking (ed.) , ''Conf. Topology of Manifolds'' , Prindle, Weber &amp; Schmidt (1968) pp. 115–133 {{MR|0242163}} {{ZBL|0179.52302}} </TD></TR></table>

Revision as of 08:01, 6 June 2020


An isotopy class of imbeddings of a sphere into a sphere. More precisely, an $ n $- dimensional knot of codimension $ q $ is a pair $ K = ( S ^ {n+} q , k ^ {n} ) $ consisting of an oriented sphere $ S ^ {n+} q $ and an oriented, locally flat, submanifold of it, $ k ^ {n} $, homeomorphic to the sphere $ S ^ {n} $. Two knots $ K _ {1} ( S ^ {n+} q , k _ {1} ^ {n} ) $ and $ K _ {2} = ( S ^ {n+} q , k _ {2} ^ {n} ) $ are called equivalent if there is an isotopy (in topology) of $ S ^ {n+} q $ which takes $ k _ {1} ^ {n} $ to $ k _ {2} ^ {n} $ while preserving the orientation. Depending on the category (Diff, PL or Top) from which the terms "submanifold" and "isotopy" in these definitions are taken, one speaks of smooth, piecewise-linear or topological multi-dimensional knots, respectively. In the smooth case $ k ^ {n} $ may have a non-standard differentiable structure. An $ n $- dimensional knot of codimension $ q $ which is isotopic to the standard imbedding is called a trivial, or unknotted, knot.

The study of multi-dimensional knots of codimension 1 is related to the Schoenflies conjecture. Every topological knot of codimension 1 is trivial. This is true for piecewise-linear and smooth knots if $ n \neq 3 , 4 $.

Piecewise-linear and topological multi-dimensional knots of codimension $ q \geq 3 $ are trivial. In the smooth case this is not so. The set of isotopy classes of smooth $ n $- dimensional knots of codimension $ q \geq 3 $ coincides, for $ n \geq 5 $, with the set $ \theta ^ {n+} q,n $ of cobordism classes of knots. (Two multi-dimensional knots $ K _ {1} = ( S ^ {n+} q , k _ {1} ^ {n} ) $ and $ K _ {2} = ( S ^ {n+} q , k _ {2} ^ {n} ) $ are called cobordant if there is a smooth $ ( n + 1 ) $- dimensional submanifold $ W \subset S ^ {n+} q \times I $ transversal to $ \partial ( S ^ {n+} q \times I ) $, where $ \partial W = ( k _ {1} ^ {n} \times 0 ) \cup ( - k _ {2} ^ {n} \times 1 ) $ and $ W $ is an $ h $- cobordism between $ k _ {1} ^ {n} \times 0 $ and $ k _ {2} ^ {n} \times 1 $.) The set $ \theta ^ {n+} q,n $ is an Abelian group with respect to the operation of connected sum. In this group the negative of the class of $ ( S ^ {n+} q , k ^ {n} ) $ is the cobordism class of $ ( - S ^ {n+} q , - k ^ {n} ) $, where the minus denotes reversal of orientation. There is a natural homomorphism $ \theta ^ {n+} q,n \rightarrow \theta ^ {n} $, where $ \theta ^ {n} $ is the group of $ n $- dimensional homotopy spheres; this homomorphism associates the differentiable structure of $ k ^ {n} $ to the knot $ ( S ^ {n+} q , k ^ {n} ) $. The kernel of this homomorphism, denoted by $ \Sigma ^ {n+} q,n $, is the set of isotopy classes of the standard sphere $ S ^ {n} $ in $ S ^ {n+} q $. If $ 2 q > n + 3 $, then $ \Sigma ^ {n+} q,n $ is trivial. If $ 2 q \geq n + 3 $ and $ ( n + 1 ) \not\equiv 0 $( $ \mathop{\rm mod} 4 $), then $ \theta ^ {n+} q,n $ and $ \Sigma ^ {n+} q,n $ are finite. When $ 2 q \leq n + 3 $ and $ ( n + 1 ) \not\equiv 0 $( $ \mathop{\rm mod} 4 $), then $ \theta ^ {n+} q,n $ and $ \Sigma ^ {n+} q,n $ are finitely-generated Abelian groups of rank 1 (see [1], [2]). The set of concordance classes of smooth imbeddings of $ S ^ {n} $ into $ S ^ {n+} q $ for $ q > 2 $ has also been calculated (see [3]).

The study of multi-dimensional knots of codimension 2, which will subsequently simply be called knots, proceeds quite similarly in all three categories (Diff, PL, Top). For $ n \geq 5 $ every topological knot may be transformed by an isotopy to a smooth knot. However, there are topological three-dimensional knots in $ S ^ {5} $ which are not equivalent, or even cobordant, to smooth knots (see [4]).

The set of isotopy classes of $ n $- dimensional knots (in each category) is an Abelian semi-group with respect to the operation of connected sum. It is known that for $ n = 1 $ every element in this semi-group is a finite sum of primes, and such a decomposition is unique.

An $ n $- dimensional knot $ K = ( S ^ {n+} 2 , k ^ {n} ) $ is trivial if and only if $ \pi _ {i} ( S ^ {n+} 2 \setminus k ^ {n} ) = \pi _ {i} ( S ^ {1} ) $ for all $ i \leq [ ( n + 1 ) / 2 ] $. An algebraic classification has been given (see [6]) of the knots $ K $ for which $ \pi _ {i} ( S ^ {n+} 2 \setminus k ^ {n} ) = \pi _ {i} ( S ^ {1} ) $, for all $ i \leq [( n+ 1)/2]- 1 $ and $ n $ odd (knots of type $ L $): For $ n \geq 5 $ the set of isotopy classes of such knots turns out to be in one-to-one correspondence with the set of $ S $- equivalence classes of the Seifert matrix. Knots of type $ L $ are important from the point of view of applications to algebraic geometry, since they contain all knots obtained by the following construction (see [15]). Let $ f ( z _ {1} \dots z _ {q+} 1 ) $ be a complex polynomial of non-zero degree having zero as an isolated singularity and let $ f ( 0) = 0 $. The intersection $ k $ of the hyperplane $ V = f ^ { - 1 } ( 0) $ with a small sphere $ S ^ {q+} 1 $ with centre at zero is a $ ( q - 2 ) $- connected $ ( 2 q - 1 ) $- dimensional manifold. The manifold $ k $ is homeomorphic to $ S ^ {2q-} 1 $ if and only if $ | \Delta ( 1) | = 1 $, where $ \Delta ( t) $ is the Alexander polynomial. In this case there thus arises a knot $ ( S ^ {2q+} 1 , k ) $. Such knots are called algebraic; they are all of type $ L $.

The exterior of a smooth knot $ K = ( S ^ {n+} 2 , k ^ {n} ) $ is the complement $ X $( of an open tubular neighbourhood) of $ k ^ {n} $ in $ S ^ {n+} 2 $. For $ n \geq 2 $, for each $ n $- dimensional knot $ K $ there is a knot $ \tau ( K) $ such that each knot with exterior diffeomorphic to the exterior of $ K $ is equivalent to either $ K $ or $ \tau ( K) $. If $ X _ {1} $, $ X _ {2} $ are the exteriors of two smooth $ n $- dimensional knots, $ n \geq 3 $, and $ \pi _ {1} ( X _ {1} ) = \pi _ {1} ( X _ {2} ) = \mathbf Z $, then the following statements are equivalent (see [7]): 1) $ X _ {1} $ and $ X _ {2} $ are diffeomorphic; and 2) the pairs $ ( X _ {1} , \partial X _ {1} ) $ and $ ( X _ {2} , \partial X _ {2} ) $ are homotopically equivalent. These results reduce the classification problem for knots to the homotopy classification of pairs $ ( X , \partial X ) $ and the solution of the question: Does the exterior determine the type of a knot, that is, does $ K = \tau ( K) $ hold? It is known that this equality holds for knots of type $ L $( see [6]) and for knots obtained by the Artin construction and the supertwisting construction (see [8]). However, two-dimensional knots have been found in $ S ^ {4} $ for which $ K \neq \tau ( K) $( see [9]).

The study of the homotopy type of the exterior of $ X $ is complicated because this exterior is not simply connected. If $ G $ is the group of the knot (that is, $ G = \pi _ {1} ( X) $), then $ G / [ G , G ] = \mathbf Z $, $ H _ {2} ( G) = 0 $, and the weight of $ G $( that is, the minimal number of elements not contained in a proper normal divisor) is equal to 1. For $ n \geq 3 $ these properties completely describe the class of groups of $ n $- dimensional knots (see [10]). The groups of one-dimensional and two-dimensional knots have a number of additional properties (see Knot theory; Two-dimensional knot).

Since $ H ^ {1} ( X ; \mathbf Z ) = \mathbf Z $, the exterior $ X $ has a unique infinite cyclic covering $ p : \widetilde{X} \rightarrow X $. The homology spaces $ H _ {*} ( \widetilde{X} ; \mathbf Z ) $ are $ \mathbf Z [ \mathbf Z ] $- modules. Their Alexander invariants are invariants of the knot. For algebraic properties of the modules $ H _ {*} ( \widetilde{X} ; \mathbf Z ) $ see [10][13].

Due to the fact that the group $ \mathbf Z $ acts without fixed points on an infinite cyclic covering, the $ ( n + 2 ) $- dimensional non-compact manifold $ \widetilde{X} $ has a number of the homological properties of compact $ ( n + 1 ) $- dimensional manifolds. In particular, for the homology of the manifold $ \widetilde{X} $ with coefficients from a field $ F $ there is a non-degenerate pairing

$$ H _ {n} ( \widetilde{X} ; F ) \otimes H _ {n+} 1- k ( \widetilde{X} ; F ) \rightarrow F ,\ k = 1 \dots n , $$

with properties resembling the pairing determined by the intersection index (in homology) in $ ( n + 1 ) $- dimensional compact manifolds. There is also a pairing

$$ T _ {k} \widetilde{X} \otimes T _ {n-} k \widetilde{X} \rightarrow \mathbf Q / \mathbf Z ,\ \ k = 1 \dots n - 1 , $$

similar to the linking coefficients (cf. Linking coefficient) in $ ( n + 1 ) $- dimensional manifolds (see [13]), where $ T _ {j} \widetilde{X} = \mathop{\rm Tors} H _ {j} ( \widetilde{X} ; \mathbf Z ) $. These homology pairings generate invariants of the homotopy type of the pair $ ( X , \partial X) $. To obtain algebraic invariants, finite-sheeted cyclic branched coverings are also used (see [14]).

The problem of classifying knots of codimension 2 up to cobordism, a coarser equivalence relation than isotopy type, has been completely solved for $ n > 1 $( see Cobordism of knots).

References

[1] A. Haefliger, "Knotted ()-spheres in -space" Ann. of Math. , 75 (1962) pp. 452–466 MR145539
[2] A. Haefliger, "Differentiable embeddings of in for " Ann. of Math. , 83 (1966) pp. 402–436 Zbl 0151.32502
[3] J. Levine, "A classification of differentiable knots" Ann. of Math. , 82 (1965) pp. 15–50 MR0180981 Zbl 0136.21102
[4] S. Cappell, J. Shaneson, "Topological knots and knot cobordism" Topology , 12 (1973) pp. 33–40 MR0321099 Zbl 0268.57006
[5] A.B. Sossinskii, "Decomposition of knots" Math. USSR Sb. , 10 (1970) pp. 139–150 Mat. Sb. , 81 : 1 (1970) pp. 145–158
[6] J. Levine, "An algebraic classification of some knots of codimension two" Comment. Math. Helv. , 45 (1970) pp. 185–198 MR0266226 Zbl 0211.55902
[7] R. Lashof, J. Shaneson, "Classification of knots in codimension two" Bull. Amer. Math. Soc. , 75 (1969) pp. 171–175 MR0242175 Zbl 0198.28701
[8] S. Cappell, "Superspinning and knot complements" J.C. Cantrell (ed.) C.H. Edwards jr. (ed.) , Topology of manifolds , Markham (1971) pp. 358–383 MR0276972 Zbl 0281.57001
[9] S. Cappell, J. Shaneson, "There exist inequivalent knots with the same complements" Ann. of Math. , 103 (1976) pp. 349–353 MR0413117
[10] M. Kervaire, "Les noeuds de dimensions supérieures" Bull. Soc. Math. France , 93 (1965) pp. 225–271 MR0189052 Zbl 0141.21201
[11] J. Levine, "Polynomial invariants of knots of codimension two" Ann. of Math. , 84 (1966) pp. 537–554 MR0200922 Zbl 0196.55905
[12] J. Levine, "Knot modules" , Knots, Groups and 3-Manifolds , Princeton Univ. Press (1975) pp. 25–34 MR0405437 Zbl 0336.57008
[13] M.Sh. Farber, "Duality in an infinite cyclic covering and even-dimensional knots" Math. USSR Izv. , 11 (1974) pp. 749–781 Izv. Akad. Nauk SSSR Ser. Mat. , 41 (1977) pp. 794–828 MR0515677 Zbl 0394.57011
[14] O.Ya. Viro, "Branched coverings of manifolds with boundary and link invariants I" Math. USSR Izv. , 7 (1973) pp. 1239–1256 Izv. Akad. Nauk SSSR Ser. Mat. , 37 (1973) pp. 1242–1258 Zbl 0295.55002
[15] J. Milnor, "Singular points of complex hypersurfaces" , Princeton Univ. Press (1968) MR0239612 Zbl 0184.48405

Comments

References

[a1] J.W. Milnor, "Infinite cyclic coverings" J. Hocking (ed.) , Conf. Topology of Manifolds , Prindle, Weber & Schmidt (1968) pp. 115–133 MR0242163 Zbl 0179.52302
How to Cite This Entry:
Multi-dimensional knot. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multi-dimensional_knot&oldid=47914
This article was adapted from an original article by M.Sh. Farber (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article