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 $ V ^ {2} $ is the Seifert manifold of a link $ L = ( S ^ {3} , l ) $ while

$$ \theta : H _ {1} ( V ; \mathbf Z ) \otimes H _ {1} ( V ; \mathbf Z ) \rightarrow \ \mathbf Z $$

is the Seifert pairing, then the symmetric bilinear form

$$ q : H _ {1} ( V ; \mathbf Z ) \otimes H _ {1} ( V ; \mathbf Z ) \rightarrow \mathbf Z $$

given by the equation

$$ q ( v _ {1} \otimes v _ {2} ) = \theta ( v _ {1} \otimes v _ {2} ) + \theta ( v _ {2} \otimes v _ {1} ) $$

is called the quadratic form of the link $ L $. The form $ q $ is described by the matrix $ M + M ^ \prime $, where $ M $ is the Seifert matrix and the prime denotes transposition. The form $ q $ itself is not an invariant of the link $ L $; however, its signature $ \sigma ( q) \in \mathbf Z $ and the Minkowski unit $ C _ {p} ( q) \in \{ - 1 , 1 \} $, where $ p $ 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 $ L $ and are denoted thus: $ \sigma ( L) = \sigma ( q) $, $ C _ {p} ( L) = C _ {p} ( q) $. The dimension $ n ( q) $ of the radical of the form $ q $ is also an invariant of $ L $. The number $ n ( L) = n ( q) + 1 $ is called the nullity of the link $ L $. One has the inequality: $ d ( L) \leq n ( L) \leq \mu ( L) $, where $ d ( L) $ is the maximum number of connected components which the Seifert manifold of the link $ L $ can have and $ \mu ( L) $ is the multiplicity, i.e. the number of components of the link $ L $.

Let $ N $ be a locally flat two-dimensional oriented submanifold of the ball $ D ^ {4} $ with $ N \cap S ^ {3} = \partial N = l $. The genus $ h ( N) $ of $ N $ can be estimated by the following inequality:

$$ 2h ( N) + \mu ( L) - \mu ( N) \geq | \sigma ( L) | + | n ( L) - \mu ( N) | , $$

where $ \mu ( N) $ is the number of components of $ N $. The lower bound for $ h ( N) $ is called the $ 4 $- genus or lower genus of $ L $. 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 $ S ^ {3} $ with values in $ \mathbf Z $ 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 $ \Sigma ^ {4} $ of the ball $ D ^ {4} $ with ramification over an oriented $ 2 $- dimensional surface $ N \subset D ^ {4} $ with $ N _ \cap S ^ {3} = \partial N = l $. In particular, the signature and Minkowski unit of a link $ L = ( S ^ {3} , l ) $ are equal to the corresponding signature and Minkowski unit of the manifold $ \Sigma ^ {4} $. The boundary $ \partial \Sigma ^ {4} = \Sigma ^ {3} $, which is a two-sheeted covering of the sphere $ S ^ {3} $ ramified over $ l $, is an invariant of $ L $. In the case of a knot, $ H _ {1} ( \Sigma ^ {3} ; \mathbf Z ) $ is a finite group. This group, as well as the form of the coefficients of the link

$$ \lambda : H _ {1} ( \Sigma ^ {3} ; \mathbf Z ) \otimes H _ {1} ( \Sigma ^ {3} ; \mathbf Z ) \rightarrow \mathbf Q / \mathbf Z , $$

defines a quadratic form of the knot in the following way. A group with a pairing, or a $ V $- group, is a pair $ ( G , \mu ) $ consisting of a finite Abelian group $ G $ and a non-degenerate symmetric bilinear form $ \mu : G \otimes G \rightarrow \mathbf Q / \mathbf Z $. Every symmetric non-degenerate integral $ ( n \times n ) $- matrix $ A $ determines a $ V $- group $ ( G , \mu ) $ as follows: the group $ G $ is generated by elements $ g _ {1} \dots g _ {n} $ with the following defining relations: $ \sum_{j=1}^ {n} a _ {ij} g _ {j} = 0 $, $ i = 1 \dots n $, where $ A = \| a _ {ij} \| $, while $ \mu ( g _ {i} , g _ {j} ) $ is congruent modulo 1 to the $ ( i , j ) $- th entry of $ A ^ {-1} $. It turns out that the $ V $- group defined in this way by the matrix $ M + M ^ \prime $ of the quadratic form of a knot is isomorphic to the $ V $- group $ ( H _ {1} ( \Sigma ^ {3} ; \mathbf Z ) , \lambda ) $ of the manifold $ \Sigma ^ {3} $( cf. [4], [9]). Numerical invariants of $ V $- 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 $ S ^ {3} $, 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 $ G _ {0} $ is coloured black and any two adjacent domains have different colours. Let $ G _ {0} \dots G _ {n} $ be all the black domains. Every double point $ x $ of the knot diagram corresponds in the following way to a certain number $ \eta ( x) \in \{ - 1 , 0 , 1 \} $. Let $ x $ be a point of the common boundary of two black domains $ G _ {i} $ and $ G _ {k} $. If $ i = k $, then $ \eta ( x ) = 0 $. If $ i \neq k $ then $ \eta ( x) = 1 $ if and only if one passes from the overpass to the underpass in the black domain in the clockwise sense; in the opposite case $ \eta ( x) = - 1 $. One can form the following $ ( n \times n ) $- matrix $ A = \| a _ {ij} \| $, where $ a _ {ii} $ is the sum of all numbers $ \eta ( x) $ corresponding to the double points $ x $ lying on the boundary of the domain $ G _ {i} $, and $ a _ {ik} $ for $ i \neq k $ is obtained by taking with opposite sign the sum of all numbers $ \eta ( x) $ where $ x $ ranges over all the common boundary points of $ G _ {i} $ and $ G _ {k} $. The form $ f = \sum _ {i,j} ^ {n} a _ {ij} x _ {i} x _ {j} $ is called the quadratic form of the knot diagram. The matrix $ A = \| a _ {ij} \| $ 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: $ Q _ {1} : A \rightarrow T ^ \prime A T $, where $ T $ is an integral unimodular matrix,

$$ Q _ {2} : A \rightarrow \ \left \| \begin{array}{cr} A & 0 \\ 0 &\pm 1 \\ \end{array} \right \| $$

and their inverses. The modulus of the determinant of $ A $ is an invariant of the knot, called the determinant of the knot. For every knot it is odd and equal to $ | \Delta ( - 1 ) | $, where $ \Delta ( t) $ is the Alexander polynomial (cf. Alexander invariants). The $ V $- 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 $ V $- group is isomorphic to the $ V $- group $ ( H _ {1} ( \Sigma ^ {3} ; \mathbf Z ) , \lambda ) $ of a two-sheeted covering of the sphere $ S ^ {3} $, ramified over the knot $ k $.

References

[1]	K. Reidemeister, "Knotentheorie" , Chelsea, reprint (1948) MR0345089 MR1522902 Zbl 0277.55007 Zbl 0005.12001 Zbl 58.1202.04 Zbl 52.0579.01
[2]	L. Goeritz, "Knoten und quadratische Formen" Math. Z. , 36 (1933) pp. 647–654 MR1545364 Zbl 0006.42201 Zbl 59.1237.02
[3]	H. Seifert, "Die Verschlingungsinvarianten der zyklischen Knotenüberlagerungen" Abh. Math. Sem. Univ. Hamburg , 11 (1935) pp. 84–101 Zbl 0011.17802 Zbl 61.0609.03
[4]	M. Kneser, D. Puppe, "Quadratische Formen und Verschlingungsinvarianten von Knoten" Math. Z. , 58 (1953) pp. 376–384 MR0056607 Zbl 0050.39801
[5]	R.C. Blanchfield, R.H. Fox, "Invariants of self-linking" Ann. of Math. , 53 (1951) pp. 556–564 MR0040658 Zbl 0045.44305
[6]	H.F. Trotter, "Homology of group systems with applications to knot theory" Ann. of Math. , 76 (1962) pp. 464–468 MR0143201 Zbl 0108.18302
[7]	K. Murasugi, "On a certain numerical invariant of link types" Trans. Amer. Math. Soc. , 117 (1965) pp. 387–422 MR0171275 Zbl 0137.17903
[8]	A. Tristram, "Some cobordism invariants for links" Proc. Cambridge Philos. Soc. , 66 (1969) pp. 251–264 MR0248854 Zbl 0191.54703
[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 Zbl 0295.55002

Comments

The radical of a quadratic form $ q $ on $ V $ is the space of all $ a \in V $ such that $ \phi ( a , b ) = 0 $ for all $ b \in V $. Here $ \phi $ is the symmetric bilinear form associated to $ q $.

The Minkowski unit of a quadratic form $ q $ is the same as the Hasse invariant of $ q $, 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) MR0515288 Zbl 0339.55004