# Multi-dimensional knot

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).

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