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).

References

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