View source for Knots and links, quadratic forms of

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k0556101.png" /> is the [[Seifert manifold|Seifert manifold]] of a [[Link|link]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k0556102.png" /> while

<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/k/k055/k055610/k0556103.png" /></td> </tr></table>

is the Seifert pairing, then the symmetric bilinear 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/k/k055/k055610/k0556104.png" /></td> </tr></table>

given by the equation

<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/k/k055/k055610/k0556105.png" /></td> </tr></table>

is called the quadratic form of the link <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k0556106.png" />. The form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k0556107.png" /> is described by the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k0556108.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k0556109.png" /> is the Seifert matrix and the prime denotes transposition. The form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561010.png" /> itself is not an invariant of the link <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561011.png" />; however, its signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561012.png" /> and the Minkowski unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561013.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561014.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561015.png" /> and are denoted thus: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561017.png" />. The dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561018.png" /> of the radical of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561019.png" /> is also an invariant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561020.png" />. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561021.png" /> is called the nullity of the link <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561022.png" />. One has the inequality: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561023.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561024.png" /> is the maximum number of connected components which the Seifert manifold of the link <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561025.png" /> can have and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561026.png" /> is the multiplicity, i.e. the number of components of the link <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561027.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561028.png" /> be a locally flat two-dimensional oriented submanifold of the ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561029.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561030.png" />. The genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561031.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561032.png" /> can be estimated by the following inequality:

<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/k/k055/k055610/k05561033.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561034.png" /> is the number of components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561035.png" />. The lower bound for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561036.png" /> is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561038.png" />-genus or lower genus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561039.png" />. 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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561040.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561041.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561042.png" /> of the ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561043.png" /> with ramification over an oriented <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561044.png" />-dimensional surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561045.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561046.png" />. In particular, the signature and Minkowski unit of a link <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561047.png" /> are equal to the corresponding signature and Minkowski unit of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561048.png" />. The boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561049.png" />, which is a two-sheeted covering of the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561050.png" /> ramified over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561051.png" />, is an invariant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561052.png" />. In the case of a knot, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561053.png" /> is a finite group. This group, as well as the form of the coefficients of the link

<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/k/k055/k055610/k05561054.png" /></td> </tr></table>

defines a quadratic form of the knot in the following way. A group with a pairing, or a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561056.png" />-group, is a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561057.png" /> consisting of a finite Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561058.png" /> and a non-degenerate symmetric bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561059.png" />. Every symmetric non-degenerate integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561060.png" />-matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561061.png" /> determines a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561062.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561063.png" /> as follows: the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561064.png" /> is generated by elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561065.png" /> with the following defining relations: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561067.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561068.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561069.png" /> is congruent modulo 1 to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561070.png" />-th entry of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561071.png" />. It turns out that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561072.png" />-group defined in this way by the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561073.png" /> of the quadratic form of a knot is isomorphic to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561074.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561075.png" /> of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561076.png" /> (cf. [[#References|[4]]], [[#References|[9]]]). Numerical invariants of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561077.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561078.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561079.png" /> is coloured black and any two adjacent domains have different colours. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561080.png" /> be all the black domains. Every double point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561081.png" /> of the knot diagram corresponds in the following way to a certain number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561082.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561083.png" /> be a point of the common boundary of two black domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561084.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561085.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561086.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561087.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561088.png" /> then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561089.png" /> if and only if one passes from the overpass to the underpass in the black domain in the clockwise sense; in the opposite case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561090.png" />. One can form the following <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561091.png" />-matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561092.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561093.png" /> is the sum of all numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561094.png" /> corresponding to the double points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561095.png" /> lying on the boundary of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561096.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561097.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561098.png" /> is obtained by taking with opposite sign the sum of all numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k05561099.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k055610100.png" /> ranges over all the common boundary points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k055610101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k055610102.png" />. The form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k055610103.png" /> is called the quadratic form of the knot diagram. The matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k055610104.png" /> 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: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k055610105.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k055610106.png" /> is an integral unimodular 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/k/k055/k055610/k055610107.png" /></td> </tr></table>

and their inverses. The modulus of the determinant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k055610108.png" /> is an invariant of the knot, called the determinant of the knot. For every knot it is odd and equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k055610109.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k055610110.png" /> is the Alexander polynomial (cf. [[Alexander invariants|Alexander invariants]]). The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k055610111.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k055610112.png" />-group is isomorphic to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k055610113.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k055610114.png" /> of a two-sheeted covering of the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k055610115.png" />, ramified over the knot <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k055610116.png" />.

====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k055610117.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k055610118.png" /> is the space of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k055610119.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k055610120.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k055610121.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k055610122.png" /> is the symmetric bilinear form associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k055610123.png" />.

The Minkowski unit of a quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k055610124.png" /> is the same as the Hasse invariant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055610/k055610125.png" />, also called Hasse–Minkowski invariant, Hasse symbol, and Hasse–Minkowski symbol. Cf. [[Hasse invariant|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>