Cobordism of knots
knot cobordism (proper bordism of knots, see Bordism)
An equivalence relation on the set of knots, weaker than the isotopy type relation. Two smooth -dimensional knots
and
are said to be cobordant if there exists a smooth oriented
-dimensional submanifold
of
, where
is homeomorphic to
and
. Here the minus sign indicates the opposite orientation. Knots cobordant to the trivial knot are called cobordant to zero, or slice knots. The set of (cobordance) equivalence classes of
-dimensional smooth knots is denoted by
. The operation of connected sum defines on
an Abelian group structure. The inverse of the knot cobordism class
is the knot cobordism class
.
For every even the group
is zero. The knot cobordism class of an odd-dimensional knot is defined by its Seifert matrix. A square integral matrix
is called cobordant to zero if it is unimodularly congruent to a matrix of the form
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where are square matrices of the same dimensions and 0 is the zero matrix. Two square matrices
and
are called cobordant if the matrix
![]() |
is cobordant to zero. A square integral matrix is called an
-matrix, where
or
, if
. The Seifert matrix of every
-dimensional knot is a
-matrix. For every
the cobordance relation is an equivalence relation on the set of all
-matrices. The set of equivalence classes is denoted by
. The operation of direct sum defines on
an Abelian group structure. One has the Levine homomorphism
which associates with the cobordism class of the knot
the cobordism class of the Seifert matrix of
. The Levine homomorphism is an isomorphism for all
. The homomorphism
is a monomorphism, its image is a subgroup of index 2 in
, consisting of the class of
-matrices
for which the signature of
is divisible by 16. The homomorphism
is an epimorphism; its kernel is non-trivial.
For a study of the structure of the groups and
and for the construction of a complete system of invariants of knot cobordism classes one makes use of the following construction. An isometric structure over a field
is a pair
consisting of a non-degenerate quadratic form
given on a finite-dimensional vector space
over
and an isometry
. An isometric structure
is called cobordant to zero if
contains a totally-isotropic subspace of half its dimension that is invariant under
. The operation of orthogonal sum of forms and direct sum of isometries defines an operation
on the set of isometric structures. Two isometric structures
and
are called cobordant if the isometric structure
is cobordant to zero. Let
be the set of cobordism classes of isometric structures
satisfying the condition
, where
is the characteristic polynomial of the isometry
. In the study of the groups
and
an important role is played by the imbeddings
and
, which are constructed as follows. Every cobordism class of
-matrices contains a non-degenerate matrix. If
is a non-degenerate
-matrix, put
,
and let
be the isometric structure whose form
has the given matrix
, while its isometry
has the matrix
. This gives a well-defined homomorphism
with
.
Let be an isometric structure on a vector space
and let
. Denote by
the
-primary component of
, i.e.
for large
. A polynomial
is called reciprocal if
for all
. For each irreducible reciprocal polynomial
denote by
the exponent, reduced modulo 2, with which
divides the characteristic polynomial
of the isometry
. For every reciprocal polynomial
irreducible over
, denote by
the signature of the restriction of
to
. For each prime number
and each reciprocal polynomial
irreducible over
, denote by
the restriction of
to
, where
is the field of
-adic numbers. Put
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where is the Hilbert symbol in
,
is the Hasse symbol and
is the rank of
. Two isometric structures
and
are cobordant if and only if
,
and
for all
and
for which these invariants are defined (cf. [3], [4]).
The compositions of the Levine homomorphism, the homomorphism and the functions
associate with every odd-dimensional knot
the numbers
,
,
. Two
-dimensional knots
and
, where
, are cobordant if and only if
![]() |
for all and
for which these invariants are defined.
is equal to the signature of the knot
(cf. Knots and links, quadratic forms of), where the sum is extended over all
of the form
, where
, and in the sum only a finite number of terms are distinct from zero.
Similarly one defines the group of locally flat or piecewise-linear knot cobordisms, denoted by and
, respectively. For all
one has an isomorphism
. The natural mapping
is an isomorphism for
, while for
it is a monomorphism with an image of index 2. This means, in particular, that there exists a non-smooth locally flat topologically three-dimensional knot in
(cf. [5]).
The theory of cobordism of knots is connected with the study of singularities of not locally flat or piecewise-linear imbeddings of codimension 2. If is an
-dimensional oriented manifold, imbedded as a subcomplex in an
-dimensional manifold
,
, and
is a small star-shaped neighbourhood of
in
, then the singularity of the imbedding of
in
at
may be measured as follows. The boundary
is an
-dimensional sphere, the orientation of which is defined by that of
;
is an
-dimensional sphere the orientation of which is defined by that of
. This defines an
-dimensional knot
, called the singularity of the imbedding
at the point
.
References
[1] | R.H. Fox, J.W. Milnor, "Singularities of 2-spheres in 4-space and cobordism of knots" Osaka Math. J. , 3 (1966) pp. 257–267 MR0211392 Zbl 0146.45501 |
[2] | M.A. Kervaire, "Les noeuds de dimensions supérieures" Bull. Soc. Math. France , 93 (1965) pp. 225–271 MR0189052 Zbl 0141.21201 |
[3] | J. Levine, "Knot cobordism groups in codimension 2" Comment. Math. Helv. , 44 (1969) pp. 229–244 |
[4] | J. Levine, "Invariants of knot cobordism" Invent. Math. , 8 (1969) pp. 98–110 MR0253348 Zbl 0179.52401 |
[5] | S.E. Capell, J.L. Shaneson, "Topological knots and knot cobordism" Topology , 12 (1973) pp. 33–40 MR321099 |
[6] | N.W. Stoltzfus, "Unraveling the integral knot concordance group" Mem. Amer. Math. Soc. , 12 (1977) pp. 192 MR0467764 Zbl 0366.57005 |
Comments
Another term for cobordance of knots is concordance of knots, and correspondingly one has the knot concordance group.
References
[a1] | L.H. Kaufmann, "On knots" , Princeton Univ. Press (1987) |
Cobordism of knots. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cobordism_of_knots&oldid=14400