Doubly slice knots and the Casson-Gordon invariants

Ruberman, Daniel

doi:10.1090/S0002-9947-1983-0709569-5

Doubly slice knots and the Casson-Gordon invariants
HTML articles powered by AMS MathViewer

by Daniel Ruberman

Trans. Amer. Math. Soc. 279 (1983), 569-588

DOI: https://doi.org/10.1090/S0002-9947-1983-0709569-5

PDF | Request permission

Abstract:

We find knots in all dimensions which are algebraically but not geometrically doubly slice. Our new obstructions involve the Casson-Gordon invariants of the finite cyclic covers in odd dimensions and of the infinite cyclic cover in even dimensions. These same invariants provide new criteria for amphicheirality and invertibility of even-dimensional knots.

References

Selman Akbulut and Robion Kirby, Branched covers of surfaces in $4$-manifolds, Math. Ann. 252 (1979/80), no. 2, 111–131. MR 593626, DOI 10.1007/BF01420118
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
William Browder, Surgery on simply-connected manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 65, Springer-Verlag, New York-Heidelberg, 1972. MR 0358813

Cobordism of classical knots

A. J. Casson and C. McA. Gordon, On slice knots in dimension three, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 39–53. MR 520521
M. Š. Farber, Duality in an infinite cyclic covering, and even-dimensional knots, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 4, 794–828, 959 (Russian). MR 0515677
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
Patrick M. Gilmer, Configurations of surfaces in $4$-manifolds, Trans. Amer. Math. Soc. 264 (1981), no. 2, 353–380. MR 603768, DOI 10.1090/S0002-9947-1981-0603768-7
Patrick M. Gilmer and Charles Livingston, On embedding $3$-manifolds in $4$-space, Topology 22 (1983), no. 3, 241–252. MR 710099, DOI 10.1016/0040-9383(83)90011-3
W. Hantzsche, Einlagerung von Mannigfaltigkeiten in euklidische Räume, Math. Z. 43 (1938), no. 1, 38–58 (German). MR 1545714, DOI 10.1007/BF01181085
Jonathan A. Hillman, Finite knot modules and the factorization of certain simple knots, Math. Ann. 257 (1981), no. 2, 261–274. MR 634467, DOI 10.1007/BF01458289
L. R. Hitt, Examples of higher-dimensional slice knots which are not ribbon knots, Proc. Amer. Math. Soc. 77 (1979), no. 2, 291–297. MR 542100, DOI 10.1090/S0002-9939-1979-0542100-0
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
C. Kearton, Simple knots which are doubly-null-cobordant, Proc. Amer. Math. Soc. 52 (1975), 471–472. MR 380817, DOI 10.1090/S0002-9939-1975-0380817-8
C. Kearton, An algebraic classification of some even-dimensional knots, Topology 15 (1976), no. 4, 363–373. MR 442948, DOI 10.1016/0040-9383(76)90030-6
Michel A. Kervaire, Les nœuds de dimensions supérieures, Bull. Soc. Math. France 93 (1965), 225–271 (French). MR 189052
Jerome Levine, Knot modules. I, Trans. Amer. Math. Soc. 229 (1977), 1–50. MR 461518, DOI 10.1090/S0002-9947-1977-0461518-0
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

On the homotopy and topological invariance of some rational Pontrjagin classes

6

Daniel Ruberman, Imbedding punctured lens spaces and connected sums, Pacific J. Math. 113 (1984), no. 2, 481–491. MR 749551
Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto-London, 1966. MR 0210112
Neal W. Stoltzfus, Isometries of inner product spaces and their geometric applications, Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977) Academic Press, New York-London, 1979, pp. 527–541. MR 537748
Neal W. Stoltzfus, Algebraic computations of the integral concordance and double null concordance group of knots, Knot theory (Proc. Sem., Plans-sur-Bex, 1977) Lecture Notes in Math., vol. 685, Springer, Berlin, 1978, pp. 274–290. MR 521738
D. W. Sumners, Invertible knot cobordisms, Comment. Math. Helv. 46 (1971), 240–256. MR 290351, DOI 10.1007/BF02566842
D. W. Sumners, Invertible knot cobordisms, Comment. Math. Helv. 46 (1971), 240–256. MR 290351, DOI 10.1007/BF02566842
Emery Thomas and John Wood, On manifolds representing homology classes in codimension $2$, Invent. Math. 25 (1974), 63–89. MR 383438, DOI 10.1007/BF01389998
Takaaki Yanagawa, On ribbon $2$-knots. The $3$-manifold bounded by the $2$-knots, Osaka Math. J. 6 (1969), 447–464. MR 266193
Don Bernard Zagier, Equivariant Pontrjagin classes and applications to orbit spaces. Applications of the $G$-signature theorem to transformation groups, symmetric products and number theory, Lecture Notes in Mathematics, Vol. 290, Springer-Verlag, Berlin-New York, 1972. MR 0339202
E. C. Zeeman, Twisting spun knots, Trans. Amer. Math. Soc. 115 (1965), 471–495. MR 195085, DOI 10.1090/S0002-9947-1965-0195085-8