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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} $.

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|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.