Multi-dimensional knot

An isotopy class of imbeddings of a sphere into a sphere. More precisely, an -dimensional knot of codimension is a pair consisting of an oriented sphere and an oriented, locally flat, submanifold of it, , homeomorphic to the sphere . Two knots and are called equivalent if there is an isotopy (in topology) of which takes to 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 may have a non-standard differentiable structure. An -dimensional knot of codimension 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 .

Piecewise-linear and topological multi-dimensional knots of codimension are trivial. In the smooth case this is not so. The set of isotopy classes of smooth -dimensional knots of codimension coincides, for , with the set of cobordism classes of knots. (Two multi-dimensional knots and are called cobordant if there is a smooth -dimensional submanifold transversal to , where and is an -cobordism between and .) The set is an Abelian group with respect to the operation of connected sum. In this group the negative of the class of is the cobordism class of , where the minus denotes reversal of orientation. There is a natural homomorphism , where is the group of -dimensional homotopy spheres; this homomorphism associates the differentiable structure of to the knot . The kernel of this homomorphism, denoted by , is the set of isotopy classes of the standard sphere in . If , then is trivial. If and (), then and are finite. When and (), then and are finitely-generated Abelian groups of rank 1 (see [1], [2]). The set of concordance classes of smooth imbeddings of into for 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 every topological knot may be transformed by an isotopy to a smooth knot. However, there are topological three-dimensional knots in which are not equivalent, or even cobordant, to smooth knots (see [4]).

The set of isotopy classes of -dimensional knots (in each category) is an Abelian semi-group with respect to the operation of connected sum. It is known that for every element in this semi-group is a finite sum of primes, and such a decomposition is unique.

An -dimensional knot is trivial if and only if for all . An algebraic classification has been given (see [6]) of the knots for which , for all and odd (knots of type ): For the set of isotopy classes of such knots turns out to be in one-to-one correspondence with the set of -equivalence classes of the Seifert matrix. Knots of type 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 be a complex polynomial of non-zero degree having zero as an isolated singularity and let . The intersection of the hyperplane with a small sphere with centre at zero is a -connected -dimensional manifold. The manifold is homeomorphic to if and only if , where is the Alexander polynomial. In this case there thus arises a knot . Such knots are called algebraic; they are all of type .

The exterior of a smooth knot is the complement (of an open tubular neighbourhood) of in . For , for each -dimensional knot there is a knot such that each knot with exterior diffeomorphic to the exterior of is equivalent to either or . If , are the exteriors of two smooth -dimensional knots, , and , then the following statements are equivalent (see [7]): 1) and are diffeomorphic; and 2) the pairs and are homotopically equivalent. These results reduce the classification problem for knots to the homotopy classification of pairs and the solution of the question: Does the exterior determine the type of a knot, that is, does hold? It is known that this equality holds for knots of type (see [6]) and for knots obtained by the Artin construction and the supertwisting construction (see [8]). However, two-dimensional knots have been found in for which (see [9]).

The study of the homotopy type of the exterior of is complicated because this exterior is not simply connected. If is the group of the knot (that is, ), then , , and the weight of (that is, the minimal number of elements not contained in a proper normal divisor) is equal to 1. For these properties completely describe the class of groups of -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 , the exterior has a unique infinite cyclic covering . The homology spaces are -modules. Their Alexander invariants are invariants of the knot. For algebraic properties of the modules see [10]–[13].

Due to the fact that the group acts without fixed points on an infinite cyclic covering, the -dimensional non-compact manifold has a number of the homological properties of compact -dimensional manifolds. In particular, for the homology of the manifold with coefficients from a field there is a non-degenerate pairing

with properties resembling the pairing determined by the intersection index (in homology) in -dimensional compact manifolds. There is also a pairing

similar to the linking coefficients (cf. Linking coefficient) in -dimensional manifolds (see [13]), where . These homology pairings generate invariants of the homotopy type of the pair . 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 (see Cobordism of knots).

References

[1]	A. Haefliger, "Knotted ()-spheres in -space" Ann. of Math. , 75 (1962) pp. 452–466
[2]	A. Haefliger, "Differentiable embeddings of in for " Ann. of Math. , 83 (1966) pp. 402–436
[3]	J. Levine, "A classification of differentiable knots" Ann. of Math. , 82 (1965) pp. 15–50
[4]	S. Cappell, J. Shaneson, "Topological knots and knot cobordism" Topology , 12 (1973) pp. 33–40
[5]	A.B. Sossinskii, "Decomposition of knots" Math. USSR Sb. , 10 (1970) pp. 139–150 Mat. Sb. , 81 : 1 (1970) pp. 145–158
[6]	J. Levine, "An algebraic classification of some knots of codimension two" Comment. Math. Helv. , 45 (1970) pp. 185–198
[7]	R. Lashof, J. Shaneson, "Classification of knots in codimension two" Bull. Amer. Math. Soc. , 75 (1969) pp. 171–175
[8]	S. Cappell, "Superspinning and knot complements" J.C. Cantrell (ed.) C.H. Edwards jr. (ed.) , Topology of manifolds , Markham (1971) pp. 358–383
[9]	S. Cappell, J. Shaneson, "There exist inequivalent knots with the same complements" Ann. of Math. , 103 (1976) pp. 349–353
[10]	M. Kervaire, "Les noeuds de dimensions supérieures" Bull. Soc. Math. France , 93 (1965) pp. 225–271
[11]	J. Levine, "Polynomial invariants of knots of codimension two" Ann. of Math. , 84 (1966) pp. 537–554
[12]	J. Levine, "Knot modules" , Knots, Groups and 3-Manifolds , Princeton Univ. Press (1975) pp. 25–34
[13]	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
[14]	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
[15]	J. Milnor, "Singular points of complex hypersurfaces" , Princeton Univ. Press (1968)

References

[a1]	J.W. Milnor, "Infinite cyclic coverings" J. Hocking (ed.) , Conf. Topology of Manifolds , Prindle, Weber & Schmidt (1968) pp. 115–133