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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|Seifert matrix]]). If  $  V  ^ {2} $
is the [[Seifert manifold|Seifert manifold]] of a [[Link|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|Alternating knots and links]]) is equal to its genus and coincides with half the degree of the Alexander polynomial (cf. [[Alexander invariants|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. [[#References|[4]]], [[#References|[9]]]). Numerical invariants of  $  V $-
groups may be found by the Blanchfield–Fox method [[#References|[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|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====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> K. Reidemeister, "Knotentheorie" , Chelsea, reprint (1948) {{MR|0345089}} {{MR|1522902}} {{ZBL|0277.55007}} {{ZBL|0005.12001}} {{ZBL|58.1202.04}} {{ZBL|52.0579.01}} </TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top"> L. Goeritz, "Knoten und quadratische Formen" ''Math. Z.'' , '''36''' (1933) pp. 647–654 {{MR|1545364}} {{ZBL|0006.42201}} {{ZBL|59.1237.02}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> H. Seifert, "Die Verschlingungsinvarianten der zyklischen Knotenüberlagerungen" ''Abh. Math. Sem. Univ. Hamburg'' , '''11''' (1935) pp. 84–101 {{MR|}} {{ZBL|0011.17802}} {{ZBL|61.0609.03}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> M. Kneser, D. Puppe, "Quadratische Formen und Verschlingungsinvarianten von Knoten" ''Math. Z.'' , '''58''' (1953) pp. 376–384 {{MR|0056607}} {{ZBL|0050.39801}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> R.C. Blanchfield, R.H. Fox, "Invariants of self-linking" ''Ann. of Math.'' , '''53''' (1951) pp. 556–564 {{MR|0040658}} {{ZBL|0045.44305}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> H.F. Trotter, "Homology of group systems with applications to knot theory" ''Ann. of Math.'' , '''76''' (1962) pp. 464–468 {{MR|0143201}} {{ZBL|0108.18302}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> K. Murasugi, "On a certain numerical invariant of link types" ''Trans. Amer. Math. Soc.'' , '''117''' (1965) pp. 387–422 {{MR|0171275}} {{ZBL|0137.17903}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> A. Tristram, "Some cobordism invariants for links" ''Proc. Cambridge Philos. Soc.'' , '''66''' (1969) pp. 251–264 {{MR|0248854}} {{ZBL|0191.54703}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> 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 {{MR|}} {{ZBL|0295.55002}} </TD></TR>
</table>

====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====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> D. Rolfsen, "Knots and links", Publish or Perish (1976) {{MR|0515288}} {{ZBL|0339.55004}} </TD></TR>
</table>