Knots and links, quadratic forms of

Forms associated with three-dimensional knots and links; certain invariants of these forms are topological invariants of the isotopy type of the knots and links. Quadratic forms of knots and links arise as a result of symmetrizing the Seifert pairing (cf. Seifert matrix). If is the Seifert manifold of a link while

is the Seifert pairing, then the symmetric bilinear form

given by the equation

is called the quadratic form of the link . The form is described by the matrix , where is the Seifert matrix and the prime denotes transposition. The form itself is not an invariant of the link ; however, its signature and the Minkowski unit , where is a prime number, do not depend on the choice of the Seifert manifold. They are called, respectively, the signature and the Minkowski unit of the link and are denoted thus: , . The dimension of the radical of the form is also an invariant of . The number is called the nullity of the link . One has the inequality: , where is the maximum number of connected components which the Seifert manifold of the link can have and is the multiplicity, i.e. the number of components of the link .

Let be a locally flat two-dimensional oriented submanifold of the ball with . The genus of can be estimated by the following inequality:

where is the number of components of . The lower bound for is called the -genus or lower genus of . The task of calculating the lower genus of various links is closely connected with the problem of realizing two-dimensional homology classes of four-dimensional manifolds by closed oriented surfaces of least possible genus. The lower genus of every special alternating knot (cf. Alternating knots and links) is equal to its genus and coincides with half the degree of the Alexander polynomial (cf. Alexander invariants). A slice knot (cf. Cobordism of knots) is a knot of lower genus zero. The signature and Minkowski unit of a knot are determined by its cobordism class. The function on the cobordism group of one-dimensional knots in with values in that maps the cobordism class to the signature of a representative knot is a homomorphism whose image is the subgroup of even integers. The number of knottings of a knot is not less than half its signature.

Quadratic forms of knots and links are closely connected with the two-sheeted ramified coverings of the ball with ramification over an oriented -dimensional surface with . In particular, the signature and Minkowski unit of a link are equal to the corresponding signature and Minkowski unit of the manifold . The boundary , which is a two-sheeted covering of the sphere ramified over , is an invariant of . In the case of a knot, is a finite group. This group, as well as the form of the coefficients of the link

defines a quadratic form of the knot in the following way. A group with a pairing, or a -group, is a pair consisting of a finite Abelian group and a non-degenerate symmetric bilinear form . Every symmetric non-degenerate integral -matrix determines a -group as follows: the group is generated by elements with the following defining relations: , , where , while is congruent modulo 1 to the -th entry of . It turns out that the -group defined in this way by the matrix of the quadratic form of a knot is isomorphic to the -group of the manifold (cf. [4], [9]). Numerical invariants of -groups may be found by the Blanchfield–Fox method [5]. With their help one can in certain cases find different knots having isomorphic groups.

The invariants of the link of a two-sheeted covering of , ramified over a knot, may be obtained at once from the projection of the knot by means of the following construction, which leads to a quadratic form of the diagram of the knot. The regular projection of a knot divides the plane into domains which may, in a unique way, be coloured black and white such that the infinite domain is coloured black and any two adjacent domains have different colours. Let be all the black domains. Every double point of the knot diagram corresponds in the following way to a certain number . Let be a point of the common boundary of two black domains and . If , then . If then if and only if one passes from the overpass to the underpass in the black domain in the clockwise sense; in the opposite case . One can form the following -matrix , where is the sum of all numbers corresponding to the double points lying on the boundary of the domain , and for is obtained by taking with opposite sign the sum of all numbers where ranges over all the common boundary points of and . The form is called the quadratic form of the knot diagram. The matrix is determined by the type of the knot up to the following connectedness relation: Two square matrices are said to be connected if one can pass from one to the other by a finite succession of the following operations: , where is an integral unimodular matrix,

and their inverses. The modulus of the determinant of is an invariant of the knot, called the determinant of the knot. For every knot it is odd and equal to , where is the Alexander polynomial (cf. Alexander invariants). The -group defined in the manner described above by the matrix of a quadratic form of any diagram is an invariant of the knot. Moreover, this -group is isomorphic to the -group of a two-sheeted covering of the sphere , ramified over the knot .

References

[1]	K. Reidemeister, "Knotentheorie" , Chelsea, reprint (1948)
[2]	L. Goeritz, "Knoten und quadratische Formen" Math. Z. , 36 (1933) pp. 647–654
[3]	H. Seifert, "Die Verschlingungsinvarianten der zyklischen Knotenüberlagerungen" Abh. Math. Sem. Univ. Hamburg , 11 (1935) pp. 84–101
[4]	M. Kneser, D. Puppe, "Quadratische Formen und Verschlingungsinvarianten von Knoten" Math. Z. , 58 (1953) pp. 376–384
[5]	R.C. Blanchfield, R.H. Fox, "Invariants of self-linking" Ann. of Math. , 53 (1951) pp. 556–564
[6]	H.F. Trotter, "Homology of group systems with applications to knot theory" Ann. of Math. , 76 (1962) pp. 464–468
[7]	K. Murasugi, "On a certain numerical invariant of link types" Trans. Amer. Math. Soc. , 117 (1965) pp. 387–422
[8]	A. Tristram, "Some cobordism invariants for links" Proc. Cambridge Philos. Soc. , 66 (1969) pp. 251–264
[9]	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

Comments

The radical of a quadratic form on is the space of all such that for all . Here is the symmetric bilinear form associated to .

The Minkowski unit of a quadratic form is the same as the Hasse invariant of , also called Hasse–Minkowski invariant, Hasse symbol, and Hasse–Minkowski symbol. Cf. Hasse invariant for its definition.

References

[a1]	D. Rolfsen, "Knots and links" , Publish or Perish (1976)