View source for Cobordism of knots

''knot cobordism (proper bordism of knots, see [[Bordism|Bordism]])''

An equivalence relation on the set of knots, weaker than the isotopy type relation. Two smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c0228001.png" />-dimensional knots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c0228002.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c0228003.png" /> are said to be cobordant if there exists a smooth oriented <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c0228004.png" />-dimensional submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c0228005.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c0228006.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c0228007.png" /> is homeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c0228008.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c0228009.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280010.png" />-dimensional smooth knots is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280011.png" />. The operation of connected sum defines on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280012.png" /> an Abelian group structure. The inverse of the knot cobordism class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280013.png" /> is the knot cobordism class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280014.png" />.

For every even <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280015.png" /> the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280016.png" /> is zero. The knot cobordism class of an odd-dimensional knot is defined by its [[Seifert matrix|Seifert matrix]]. A square integral matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280017.png" /> is called cobordant to zero if it is unimodularly congruent to a matrix of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280018.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280019.png" /> are square matrices of the same dimensions and 0 is the zero matrix. Two square matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280021.png" /> are called cobordant if the matrix

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280022.png" /></td> </tr></table>

is cobordant to zero. A square integral matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280023.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280025.png" />-matrix, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280026.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280027.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280028.png" />. The Seifert matrix of every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280029.png" />-dimensional knot is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280030.png" />-matrix. For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280031.png" /> the cobordance relation is an equivalence relation on the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280032.png" />-matrices. The set of equivalence classes is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280033.png" />. The operation of direct sum defines on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280034.png" /> an Abelian group structure. One has the Levine homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280035.png" /> which associates with the cobordism class of the knot <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280036.png" /> the cobordism class of the Seifert matrix of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280037.png" />. The Levine homomorphism is an isomorphism for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280038.png" />. The homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280039.png" /> is a monomorphism, its image is a subgroup of index 2 in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280040.png" />, consisting of the class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280041.png" />-matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280042.png" /> for which the signature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280043.png" /> is divisible by 16. The homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280044.png" /> is an epimorphism; its kernel is non-trivial.

For a study of the structure of the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280046.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280047.png" /> is a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280048.png" /> consisting of a non-degenerate quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280049.png" /> given on a finite-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280050.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280051.png" /> and an isometry <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280052.png" />. An isometric structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280053.png" /> is called cobordant to zero if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280054.png" /> contains a totally-isotropic subspace of half its dimension that is invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280055.png" />. The operation of orthogonal sum of forms and direct sum of isometries defines an operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280056.png" /> on the set of isometric structures. Two isometric structures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280058.png" /> are called cobordant if the isometric structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280059.png" /> is cobordant to zero. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280060.png" /> be the set of cobordism classes of isometric structures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280061.png" /> satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280062.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280063.png" /> is the characteristic polynomial of the isometry <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280064.png" />. In the study of the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280066.png" /> an important role is played by the imbeddings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280067.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280068.png" />, which are constructed as follows. Every cobordism class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280069.png" />-matrices contains a non-degenerate matrix. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280070.png" /> is a non-degenerate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280071.png" />-matrix, put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280073.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280074.png" /> be the isometric structure whose form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280075.png" /> has the given matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280076.png" />, while its isometry <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280077.png" /> has the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280078.png" />. This gives a well-defined homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280079.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280080.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280081.png" /> be an isometric structure on a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280082.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280083.png" />. Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280084.png" /> the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280086.png" />-primary component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280087.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280088.png" /> for large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280089.png" />. A polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280090.png" /> is called reciprocal if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280091.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280092.png" />. For each irreducible reciprocal polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280093.png" /> denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280094.png" /> the exponent, reduced modulo 2, with which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280095.png" /> divides the characteristic polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280096.png" /> of the isometry <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280097.png" />. For every reciprocal polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280098.png" /> irreducible over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c02280099.png" />, denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800100.png" /> the signature of the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800101.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800102.png" />. For each prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800103.png" /> and each reciprocal polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800104.png" /> irreducible over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800105.png" />, denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800106.png" /> the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800107.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800108.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800109.png" /> is the field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800110.png" />-adic numbers. Put

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800111.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800112.png" /> is the Hilbert symbol in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800113.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800114.png" /> is the Hasse symbol and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800115.png" /> is the rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800116.png" />. Two isometric structures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800117.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800118.png" /> are cobordant if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800119.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800120.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800121.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800122.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800123.png" /> for which these invariants are defined (cf. [[#References|[3]]], [[#References|[4]]]).

The compositions of the Levine homomorphism, the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800124.png" /> and the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800125.png" /> associate with every odd-dimensional knot <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800126.png" /> the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800127.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800128.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800129.png" />. Two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800130.png" />-dimensional knots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800131.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800132.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800133.png" />, are cobordant if and only if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800134.png" /></td> </tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800135.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800136.png" /> for which these invariants are defined. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800137.png" /> is equal to the signature of the knot <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800138.png" /> (cf. [[Knots and links, quadratic forms of|Knots and links, quadratic forms of]]), where the sum is extended over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800139.png" /> of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800140.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800141.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800142.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800143.png" />, respectively. For all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800144.png" /> one has an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800145.png" />. The natural mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800146.png" /> is an isomorphism for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800147.png" />, while for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800148.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800149.png" /> (cf. [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800150.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800151.png" />-dimensional oriented manifold, imbedded as a subcomplex in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800152.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800153.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800154.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800155.png" /> is a small star-shaped neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800156.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800157.png" />, then the singularity of the imbedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800158.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800159.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800160.png" /> may be measured as follows. The boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800161.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800162.png" />-dimensional sphere, the orientation of which is defined by that of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800163.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800164.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800165.png" />-dimensional sphere the orientation of which is defined by that of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800166.png" />. This defines an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800167.png" />-dimensional knot <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800168.png" />, called the singularity of the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800169.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022800/c022800170.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.A. Kervaire,   "Les noeuds de dimensions supérieures"  ''Bull. Soc. Math. France'' , '''93'''  (1965)  pp. 225–271</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Levine,   "Knot cobordism groups in codimension 2"  ''Comment. Math. Helv.'' , '''44'''  (1969)  pp. 229–244</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J. Levine,   "Invariants of knot cobordism"  ''Invent. Math.'' , '''8'''  (1969)  pp. 98–110</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S.E. Capell,   J.L. Shaneson,   "Topological knots and knot cobordism"  ''Topology'' , '''12'''  (1973)  pp. 33–40</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  N.W. Stoltzfus,   "Unraveling the integral knot concordance group"  ''Mem. Amer. Math. Soc.'' , '''12'''  (1977)  pp. 192</TD></TR></table>



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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.H. Kaufmann,   "On knots" , Princeton Univ. Press  (1987)</TD></TR></table>