# Cobordism of knots

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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 $x$.

#### 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