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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 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 $.

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| $  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]]]).

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 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 [[#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 $.

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]]]).

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]]).

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]]].

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)|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|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====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> A. Haefliger, "Knotted ($4k-1$)-spheres in $6k$-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 $S^n$ in $S^{n+q}$ for $q>2$" ''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>
<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>