Signature of links

Kauffman, Louis H.; Taylor, Laurence R.

doi:10.1090/S0002-9947-1976-0388373-0

Signature of links
HTML articles powered by AMS MathViewer

by Louis H. Kauffman and Laurence R. Taylor PDF

Trans. Amer. Math. Soc. 216 (1976), 351-365 Request permission

Abstract:

Let L be an oriented tame link in the three sphere ${S^3}$. We study the Murasugi signature, $\sigma (L)$, and the nullity, $\eta (L)$. It is shown that $\sigma (L)$ is a locally flat topological concordance invariant and that $\eta (L)$ is a topological concordance invariant (no local flatness assumption here). Known results about the signature are re-proved (in some cases generalized) using branched coverings.

References

M. F. Atiyah and I. M. Singer, The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546–604. MR 236952, DOI 10.2307/1970717
Dieter Erle, Die quadratische Form eines Knotens und ein Satz über Knotenmannigfaltigkeiten, J. Reine Angew. Math. 236 (1969), 174–218 (German). MR 248856, DOI 10.1515/crll.1969.236.174
R. H. Fox, A quick trip through knot theory, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 120–167. MR 0140099
Fujitsugu Hosokawa, On $\nabla$-polynomials of links, Osaka Math. J. 10 (1958), 273–282. MR 102820
Fujitsugu Hosokawa and Shin’ichi Kinoshita, On the homology group of branched cyclic covering spaces of links, Osaka Math. J. 12 (1960), 331–355. MR 125579
K. Jänich and E. Ossa, On the signature of an involution, Topology 8 (1969), 27–30. MR 238337, DOI 10.1016/0040-9383(69)90027-5

Cyclic branched covers

actions, and hypersurface singularities

Louis H. Kauffman, Branched coverings, open books and knot periodicity, Topology 13 (1974), 143–160. MR 375337, DOI 10.1016/0040-9383(74)90005-6

Codimension two locally flat imbeddings

18

Hellmuth Kneser, Die Deformationssätze der einfach zusammenhängenden Flächen, Math. Z. 25 (1926), no. 1, 362–372 (German). MR 1544816, DOI 10.1007/BF01283844
John W. Milnor, Infinite cyclic coverings, Conference on the Topology of Manifolds (Michigan State Univ., E. Lansing, Mich., 1967) Prindle, Weber & Schmidt, Boston, Mass., 1968, pp. 115–133. MR 0242163
John Milnor, Link groups, Ann. of Math. (2) 59 (1954), 177–195. MR 71020, DOI 10.2307/1969685
John Milnor, Isotopy of links, Algebraic geometry and topology. A symposium in honor of S. Lefschetz, Princeton University Press, Princeton, N.J., 1957, pp. 280–306. MR 0092150
Kunio Murasugi, On a certain numerical invariant of link types, Trans. Amer. Math. Soc. 117 (1965), 387–422. MR 171275, DOI 10.1090/S0002-9947-1965-0171275-5
Kunio Murasugi, On the signature of links, Topology 9 (1970), 283–298. MR 261585, DOI 10.1016/0040-9383(70)90018-2
H. Seifert, Über das Geschlecht von Knoten, Math. Ann. 110 (1935), no. 1, 571–592 (German). MR 1512955, DOI 10.1007/BF01448044
Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112

On the homology class of a topological hypersurface

René Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86 (French). MR 61823, DOI 10.1007/BF02566923
H. F. Trotter, Homology of group systems with applications to knot theory, Ann. of Math. (2) 76 (1962), 464–498. MR 143201, DOI 10.2307/1970369
C. T. C. Wall, Diffeomorphisms of $4$-manifolds, J. London Math. Soc. 39 (1964), 131–140. MR 163323, DOI 10.1112/jlms/s1-39.1.131

A certain region in Euclidean

space

21