Two-dimensional knot

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.