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 $ n $- dimensional knots $ K _ {1} = ( S ^ {n + 2 } , k _ {1} ^ {n} ) $ and $ K _ {2} = ( S ^ {n + 2 } , k _ {2} ^ {n} ) $ are said to be cobordant if there exists a smooth oriented $ ( n + 1 ) $- dimensional submanifold $ V $ of $ [ 0, 1] \times S ^ {n + 2 } $, where $ V $ is homeomorphic to $ [ 0, 1] \times S ^ {n} $ and $ \partial V = V \cap \{ 0, 1 \} \times ( S ^ {n + 2 } ) = ( 0 \times k _ {1} ) \cup ( 1 \times - k _ {2} ) $. 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 $ n $- dimensional smooth knots is denoted by $ C _ {n} $. The operation of connected sum defines on $ C _ {n} $ an Abelian group structure. The inverse of the knot cobordism class $ ( S ^ {n + 2 } , k ^ {n} ) $ is the knot cobordism class $ (- S ^ {n + 2 } , - k ^ {n} ) $.

For every even $ n $ the group $ C _ {n} $ is zero. The knot cobordism class of an odd-dimensional knot is defined by its Seifert matrix. A square integral matrix $ A $ is called cobordant to zero if it is unimodularly congruent to a matrix of the form

$$ \left \| \begin{array}{ll} 0 &N _ {1} \\ N _ {2} &N _ {3} \\ \end{array} \right \| , $$

where $ N _ {1} , N _ {2} , N _ {3} $ are square matrices of the same dimensions and 0 is the zero matrix. Two square matrices $ A _ {1} $ and $ A _ {2} $ are called cobordant if the matrix

$$ \left \| \begin{array}{lr} A _ {1} & 0 \\ 0 &- A _ {2} \\ \end{array} \ \right \| $$

is cobordant to zero. A square integral matrix $ A $ is called an $ \epsilon $- matrix, where $ \epsilon = + 1 $ or $ - 1 $, if $ \mathop{\rm det} ( A + \epsilon A ^ \prime ) = \pm 1 $. The Seifert matrix of every $ ( 2q - 1) $- dimensional knot is a $ (- 1) ^ {q} $- matrix. For every $ \epsilon $ the cobordance relation is an equivalence relation on the set of all $ \epsilon $- matrices. The set of equivalence classes is denoted by $ G _ \epsilon $. The operation of direct sum defines on $ G _ \epsilon $ an Abelian group structure. One has the Levine homomorphism $ \phi _ {q} : C _ {2q - 1 } \rightarrow G _ {(- 1) ^ {q} } $ which associates with the cobordism class of the knot $ K $ the cobordism class of the Seifert matrix of $ K $. The Levine homomorphism is an isomorphism for all $ q \geq 3 $. The homomorphism $ \phi _ {2} : C _ {3} \rightarrow G _ {+} 1 $ is a monomorphism, its image is a subgroup of index 2 in $ G _ {+} 1 $, consisting of the class of $ (+ 1) $- matrices $ A $ for which the signature of $ A + A ^ \prime $ is divisible by 16. The homomorphism $ \phi _ {1} : C _ {+} 1 \rightarrow G _ {-} 1 $ is an epimorphism; its kernel is non-trivial.

For a study of the structure of the groups $ G _ {+} 1 $ and $ G _ {-} 1 $ 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 $ F $ is a pair $ (\langle , \rangle; T) $ consisting of a non-degenerate quadratic form $ \langle , \rangle $ given on a finite-dimensional vector space $ V $ over $ F $ and an isometry $ T: V \rightarrow V $. An isometric structure $ (\langle , \rangle; T) $ is called cobordant to zero if $ V $ contains a totally-isotropic subspace of half its dimension that is invariant under $ T $. The operation of orthogonal sum of forms and direct sum of isometries defines an operation $ \perp $ on the set of isometric structures. Two isometric structures $ (\langle , \rangle; T) $ and $ (\langle , \rangle ^ \prime ; T ^ { \prime } ) $ are called cobordant if the isometric structure $ (\langle , \rangle; T) \perp (- \langle , \rangle ^ \prime ; T ^ { \prime } ) $ is cobordant to zero. Let $ G _ {F} $ be the set of cobordism classes of isometric structures $ (\langle , \rangle; T) $ satisfying the condition $ \Delta _ {T} ( 1) \times \Delta _ {T} (- 1) \neq 0 $, where $ \Delta _ {T} ( t) $ is the characteristic polynomial of the isometry $ T $. In the study of the groups $ G _ {+} 1 $ and $ G _ {-} 1 $ an important role is played by the imbeddings $ \chi _ {+} : G _ {+} 1 \rightarrow G _ {Q} $ and $ \chi _ {-} : G _ {-} 1 \rightarrow G _ {Q} $, which are constructed as follows. Every cobordism class of $ \epsilon $- matrices contains a non-degenerate matrix. If $ A $ is a non-degenerate $ \epsilon $- matrix, put $ B = - A ^ {-} 1 A ^ \prime $, $ Q = A + A ^ \prime $ and let $ (\langle , \rangle; T) $ be the isometric structure whose form $ \langle , \rangle $ has the given matrix $ Q $, while its isometry $ T $ has the matrix $ B $. This gives a well-defined homomorphism $ \chi _ \epsilon $ with $ \mathop{\rm ker} \chi _ \epsilon = 0 $.

Let $ \alpha = (\langle , \rangle; T) $ be an isometric structure on a vector space $ V $ and let $ \lambda \in F [ t] $. Denote by $ V _ \lambda $ the $ \lambda $- primary component of $ V $, i.e. $ V _ \lambda = \mathop{\rm ker} \lambda ( T) ^ {N} $ for large $ N $. A polynomial $ \lambda ( t) = t ^ {k} + a _ {1} t ^ {k - 1 } + \dots + 1 $ is called reciprocal if $ a _ {i} = a _ {k - i } $ for all $ i $. For each irreducible reciprocal polynomial $ \lambda \in \mathbf Q [ t] $ denote by $ \epsilon _ \lambda ( \alpha ) $ the exponent, reduced modulo 2, with which $ \lambda $ divides the characteristic polynomial $ \Delta _ {T} $ of the isometry $ T $. For every reciprocal polynomial $ \lambda \in \mathbf R [ t] $ irreducible over $ \mathbf R [ t] $, denote by $ \sigma _ \lambda ( \alpha ) $ the signature of the restriction of $ \langle , \rangle $ to $ ( V \otimes \mathbf R ) _ \lambda $. For each prime number $ p $ and each reciprocal polynomial $ \lambda \in \mathbf Q _ {p} [ t] $ irreducible over $ \mathbf Q _ {p} [ t] $, denote by $ \langle , \rangle _ \lambda ^ {p} $ the restriction of $ \langle , \rangle $ to $ ( V \otimes \mathbf Q _ {p} ) _ \lambda $, where $ \mathbf Q _ {p} $ is the field of $ p $- adic numbers. Put

$$ \mu _ \lambda ^ {p} ( \alpha ) = \ (- 1, 1) ^ {r ( r + 3) / 2 } ( \mathop{\rm det} \langle , \rangle _ \lambda ^ {p} , - 1 ) ^ {r} S (\langle , \rangle _ \lambda ^ {p} ), $$

where $ ( , ) $ is the Hilbert symbol in $ \mathbf Q _ {p} $, $ S $ is the Hasse symbol and $ 2r $ is the rank of $ \langle , \rangle _ \lambda ^ {p} $. Two isometric structures $ \alpha $ and $ \beta $ are cobordant if and only if $ \epsilon _ \lambda ( \alpha ) = \epsilon _ \lambda ( \beta ) $, $ \sigma _ \lambda ( \alpha ) = \sigma _ \lambda ( \beta ) $ and $ \mu _ \lambda ^ {p} ( \alpha ) = \mu _ \lambda ^ {p} ( \beta ) $ for all $ \lambda $ and $ p $ for which these invariants are defined (cf. [3], [4]).

The compositions of the Levine homomorphism, the homomorphism $ \chi $ and the functions $ \epsilon _ \lambda , \sigma _ \lambda , \mu _ \lambda ^ {p} $ associate with every odd-dimensional knot $ K $ the numbers $ \epsilon _ \lambda ( K) \in \{ 0, 1 \} $, $ \sigma _ \lambda ( K) \in \mathbf Z $, $ \mu _ \lambda ^ {p} ( K) \in \{ - 1, 1 \} $. Two $ ( 2q - 1) $- dimensional knots $ K _ {1} $ and $ K _ {2} $, where $ q > 1 $, are cobordant if and only if

$$ \epsilon _ \lambda ( K _ {1} ) = \ \epsilon ( K _ {2} ),\ \ \sigma _ \lambda ( K _ {1} ) = \ \sigma _ \lambda ( K _ {2} ),\ \ \mu _ \lambda ^ {p} ( K _ {1} ) = \ \mu _ \lambda ^ {p} ( K _ {2} ) $$

for all $ \lambda $ and $ p $ for which these invariants are defined. $ \sum \sigma _ \lambda ( K) $ is equal to the signature of the knot $ K $( cf. Knots and links, quadratic forms of), where the sum is extended over all $ \lambda ( t) $ of the form $ t ^ {2} - 2t \cos \theta + 1 $, where $ 0 < \theta < \pi $, 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 $ C _ {n} ^ { \mathop{\rm TOP} } $ and $ C _ {n} ^ { \mathop{\rm PL} } $, respectively. For all $ n $ one has an isomorphism $ C _ {n} ^ { \mathop{\rm PL} } \approx C _ {n} $. The natural mapping $ C _ {n} \rightarrow C _ {n} ^ { \mathop{\rm TOP} } $ is an isomorphism for $ n > 3 $, while for $ n = 3 $ 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 $ S ^ {5} $( 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 $ P $ is an $ ( n + 2 ) $- dimensional oriented manifold, imbedded as a subcomplex in an $ ( n + 3 ) $- dimensional manifold $ M $, $ x \in P $, and $ N $ is a small star-shaped neighbourhood of $ x $ in $ M $, then the singularity of the imbedding of $ P $ in $ M $ at $ x $ may be measured as follows. The boundary $ \partial N $ is an $ ( n + 2 ) $- dimensional sphere, the orientation of which is defined by that of $ M $; $ P \cap \partial N $ is an $ n $- dimensional sphere the orientation of which is defined by that of $ P $. This defines an $ n $- dimensional knot $ ( \partial N, \partial N \cap P) $, called the singularity of the imbedding $ P \subset M $ at the point $ x $.

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Another term for cobordance of knots is concordance of knots, and correspondingly one has the knot concordance group.

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