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A class of isotopic imbeddings of the two-dimensional sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t0945201.png" /> in the four-dimensional one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t0945202.png" />. The condition of local planarity is usually imposed. The method of study consists in considering sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t0945203.png" /> by a bundle of three-dimensional parallel planes. The fundamental problem is whether or not the knot will be trivial if its group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t0945204.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t0945205.png" />. It is known that in such a case the complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t0945206.png" /> has the homotopy type of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t0945207.png" />.
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A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t0945209.png" />-ribbon in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452010.png" /> is the image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452011.png" /> of an immersion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452013.png" /> is the three-dimensional disc, such that: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452014.png" /> is an imbedding; 2) the self-intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452015.png" /> consists of a finite number of pairwise non-intersecting two-dimensional discs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452016.png" />; and 3) the pre-image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452017.png" /> of each disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452018.png" /> is a union of two discs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452020.png" /> such that
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452021.png" /></td> </tr></table>
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A class of isotopic imbeddings of the two-dimensional sphere  $  S  ^ {2} $
 +
in the four-dimensional one  $  S  ^ {4} $.
 +
The condition of local planarity is usually imposed. The method of study consists in considering sections of  $  S  ^ {2} $
 +
by a bundle of three-dimensional parallel planes. The fundamental problem is whether or not the knot will be trivial if its group  $  \pi _ {1} ( S  ^ {4} \setminus  S  ^ {2} ) $
 +
is isomorphic to  $  \mathbf Z $.  
 +
It is known that in such a case the complement  $  S  ^ {4} \setminus  S  ^ {2} $
 +
has the homotopy type of  $  S  ^ {1} $.
  
The image of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452022.png" /> is a two-dimensional knot in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452023.png" />. The knots thus obtained are said to be ribbon knots. This is one of the most thoroughly studied class of two-dimensional knots. Any two-dimensional ribbon knot is the boundary of some three-dimensional submanifold of the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452024.png" /> which is homeomorphic either to the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452025.png" /> or to the connected sum of some number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452026.png" />. A two-dimensional ribbon knot is trivial if and only if the fundamental group of its complement is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452027.png" />. A group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452028.png" /> is the group of some two-dimensional ribbon knot in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452029.png" /> if and only if it has a Wirtinger presentation (i.e. a presentation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452030.png" />, where each relation has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452031.png" />) in which the number of relations is one smaller than the number of generators and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452032.png" />.
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A  $  3 $-
 +
ribbon in $  S  ^ {4} $
 +
is the image  $  D  ^ {3} $
 +
of an immersion  $  \phi :  \Delta  ^ {3} \rightarrow S  ^ {4} $,
 +
where  $  \Delta  ^ {3} $
 +
is the three-dimensional disc, such that: 1)  $  \phi \mid  _ {\partial  \Delta  ^ {3}  } $
 +
is an imbedding; 2) the self-intersection of  $  \phi $
 +
consists of a finite number of pairwise non-intersecting two-dimensional discs  $  D _ {1} \dots D _ {n} $;
 +
and 3) the pre-image  $  \phi  ^ {-} 1 ( D _ {i} ) $
 +
of each disc  $  D _ {i} $
 +
is a union of two discs  $  D _ {i}  ^  \prime  $
 +
and $  D _ {i}  ^ {\prime\prime} $
 +
such that
  
The class of groups of all two-dimensional knots has not yet been fully described. It is known that this class is wider than that of the one-dimensional knots in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452033.png" /> but smaller than the class of groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452034.png" />-dimensional knots in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452036.png" />. The latter class has been fully characterized (cf. [[Multi-dimensional knot|Multi-dimensional knot]]). The following properties are displayed by two-dimensional knot groups (but not, in general, by the groups of three-dimensional knots in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452037.png" />):
+
$$
 +
D _ {i}  ^  \prime  \cap D _ {i}  ^ {\prime\prime}  = \emptyset ,\ \
 +
D _ {i}  ^  \prime  \subset    \mathop{\rm int}  \Delta  ^ {3} ,\ \
 +
\partial  D _ {i}  ^ {\prime\prime}  = D _ {i}  ^ {\prime\prime} \cap \partial  \Delta  ^ {3} .
 +
$$
  
<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/t/t094/t094520/t09452038.png" /></td> </tr></table>
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The image of the boundary  $  \partial  \Delta  ^ {3} $
 +
is a two-dimensional knot in  $  S  ^ {4} $.
 +
The knots thus obtained are said to be ribbon knots. This is one of the most thoroughly studied class of two-dimensional knots. Any two-dimensional ribbon knot is the boundary of some three-dimensional submanifold of the sphere  $  S  ^ {4} $
 +
which is homeomorphic either to the disc  $  \Delta  ^ {3} $
 +
or to the connected sum of some number of  $  ( S  ^ {1} \times S  ^ {2} ) \setminus  \Delta  ^ {3} $.
 +
A two-dimensional ribbon knot is trivial if and only if the fundamental group of its complement is isomorphic to  $  \mathbf Z $.
 +
A group  $  G $
 +
is the group of some two-dimensional ribbon knot in  $  S  ^ {4} $
 +
if and only if it has a Wirtinger presentation (i.e. a presentation  $  | x _ {1} \dots x _ {n} : r _ {1} \dots r _ {m} | $,
 +
where each relation has the form  $  x _ {i} = \omega _ {i , j }  x _ {j} \omega _ {i , j }  ^ {-} 1 $)
 +
in which the number of relations is one smaller than the number of generators and  $  G / [ G , G ] = \mathbf Z $.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452039.png" /> is the commutator subgroup; on the finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452040.png" /> there exists a non-degenerate symmetric form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452041.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452043.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452044.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452045.png" /> is the automorphism induced in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452046.png" /> by conjugation with the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452047.png" />.
+
The class of groups of all two-dimensional knots has not yet been fully described. It is known that this class is wider than that of the one-dimensional knots in  $  S  ^ {3} $
 +
but smaller than the class of groups of  $  k $-
 +
dimensional knots in  $  S  ^ {k+} 2 $,  
 +
$  k \geq  3 $.  
 +
The latter class has been fully characterized (cf. [[Multi-dimensional knot|Multi-dimensional knot]]). The following properties are displayed by two-dimensional knot groups (but not, in general, by the groups of three-dimensional knots in  $  S  ^ {5} $):
  
The calculation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094520/t09452048.png" /> has been done only for special types of two-dimensional knots, e.g. those obtained by Artin's construction, ribbon knots and fibred knots.
+
$$
 +
\mathop{\rm dim}  H _ {2} ( G  ^  \prime  , \mathbf Q )  \leq    \mathop{\rm dim} \
 +
H _ {1} ( G  ^  \prime  , \mathbf Q ) ,
 +
$$
 +
 
 +
where  $  G  ^  \prime  = [ G , G ] $
 +
is the commutator subgroup; on the finite group  $  T = \mathop{\rm Tors} ( G  ^  \prime  / G  ^ {\prime\prime} ) $
 +
there exists a non-degenerate symmetric form  $  L :  T \otimes T \rightarrow \mathbf Q / \mathbf Z $
 +
such that for any  $  m \in G $,
 +
$  x , y \in T $
 +
one has  $  L ( x , y ) = L ( \tau x , \tau y ) $,
 +
where  $  \tau : T \rightarrow T $
 +
is the automorphism induced in  $  G $
 +
by conjugation with the element  $  m $.
 +
 
 +
The calculation of  $  \pi _ {2} ( S  ^ {4} \setminus  S  ^ {2} ) $
 +
has been done only for special types of two-dimensional knots, e.g. those obtained by Artin's construction, ribbon knots and fibred knots.

Latest revision as of 08:26, 6 June 2020


A class of isotopic imbeddings of the two-dimensional sphere $ S ^ {2} $ in the four-dimensional one $ S ^ {4} $. The condition of local planarity is usually imposed. The method of study consists in considering sections of $ S ^ {2} $ by a bundle of three-dimensional parallel planes. The fundamental problem is whether or not the knot will be trivial if its group $ \pi _ {1} ( S ^ {4} \setminus S ^ {2} ) $ is isomorphic to $ \mathbf Z $. It is known that in such a case the complement $ S ^ {4} \setminus S ^ {2} $ has the homotopy type of $ S ^ {1} $.

A $ 3 $- ribbon in $ S ^ {4} $ is the image $ D ^ {3} $ of an immersion $ \phi : \Delta ^ {3} \rightarrow S ^ {4} $, where $ \Delta ^ {3} $ is the three-dimensional disc, such that: 1) $ \phi \mid _ {\partial \Delta ^ {3} } $ is an imbedding; 2) the self-intersection of $ \phi $ consists of a finite number of pairwise non-intersecting two-dimensional discs $ D _ {1} \dots D _ {n} $; and 3) the pre-image $ \phi ^ {-} 1 ( D _ {i} ) $ of each disc $ D _ {i} $ is a union of two discs $ D _ {i} ^ \prime $ and $ D _ {i} ^ {\prime\prime} $ such that

$$ D _ {i} ^ \prime \cap D _ {i} ^ {\prime\prime} = \emptyset ,\ \ D _ {i} ^ \prime \subset \mathop{\rm int} \Delta ^ {3} ,\ \ \partial D _ {i} ^ {\prime\prime} = D _ {i} ^ {\prime\prime} \cap \partial \Delta ^ {3} . $$

The image of the boundary $ \partial \Delta ^ {3} $ is a two-dimensional knot in $ S ^ {4} $. The knots thus obtained are said to be ribbon knots. This is one of the most thoroughly studied class of two-dimensional knots. Any two-dimensional ribbon knot is the boundary of some three-dimensional submanifold of the sphere $ S ^ {4} $ which is homeomorphic either to the disc $ \Delta ^ {3} $ or to the connected sum of some number of $ ( S ^ {1} \times S ^ {2} ) \setminus \Delta ^ {3} $. A two-dimensional ribbon knot is trivial if and only if the fundamental group of its complement is isomorphic to $ \mathbf Z $. A group $ G $ is the group of some two-dimensional ribbon knot in $ S ^ {4} $ if and only if it has a Wirtinger presentation (i.e. a presentation $ | x _ {1} \dots x _ {n} : r _ {1} \dots r _ {m} | $, where each relation has the form $ x _ {i} = \omega _ {i , j } x _ {j} \omega _ {i , j } ^ {-} 1 $) in which the number of relations is one smaller than the number of generators and $ G / [ G , G ] = \mathbf Z $.

The class of groups of all two-dimensional knots has not yet been fully described. It is known that this class is wider than that of the one-dimensional knots in $ S ^ {3} $ but smaller than the class of groups of $ k $- dimensional knots in $ S ^ {k+} 2 $, $ k \geq 3 $. The latter class has been fully characterized (cf. Multi-dimensional knot). The following properties are displayed by two-dimensional knot groups (but not, in general, by the groups of three-dimensional knots in $ S ^ {5} $):

$$ \mathop{\rm dim} H _ {2} ( G ^ \prime , \mathbf Q ) \leq \mathop{\rm dim} \ H _ {1} ( G ^ \prime , \mathbf Q ) , $$

where $ G ^ \prime = [ G , G ] $ is the commutator subgroup; on the finite group $ T = \mathop{\rm Tors} ( G ^ \prime / G ^ {\prime\prime} ) $ there exists a non-degenerate symmetric form $ L : T \otimes T \rightarrow \mathbf Q / \mathbf Z $ such that for any $ m \in G $, $ x , y \in T $ one has $ L ( x , y ) = L ( \tau x , \tau y ) $, where $ \tau : T \rightarrow T $ is the automorphism induced in $ G $ by conjugation with the element $ m $.

The calculation of $ \pi _ {2} ( S ^ {4} \setminus S ^ {2} ) $ has been done only for special types of two-dimensional knots, e.g. those obtained by Artin's construction, ribbon knots and fibred knots.

How to Cite This Entry:
Two-dimensional knot. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Two-dimensional_knot&oldid=49051
This article was adapted from an original article by A.V. ChernavskiiM.Sh. Farber (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article