An algebraic classification of certain simple even-dimensional knots

Kearton, C.

doi:10.1090/S0002-9947-1983-0684492-3

An algebraic classification of certain simple even-dimensional knots
HTML articles powered by AMS MathViewer

by C. Kearton PDF

Trans. Amer. Math. Soc. 276 (1983), 1-53 Request permission

Abstract:

The simple $2q$-knots, $q \geqslant 4$, for which ${H_q}(\tilde {K})$ contains no ${\mathbf {Z}}$-torsion, are classified by means of Hermitian duality pairings on their homology and homotopy modules.

References

Hyman Bass, Algebraic $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491
Richard C. Blanchfield, Intersection theory of manifolds with operators with applications to knot theory, Ann. of Math. (2) 65 (1957), 340–356. MR 85512, DOI 10.2307/1969966
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
M. Š. Farber, Isotopy types of knots of codimension two, Trans. Amer. Math. Soc. 261 (1980), no. 1, 185–209. MR 576871, DOI 10.1090/S0002-9947-1980-0576871-7
Sze-tsen Hu, Homotopy theory, Pure and Applied Mathematics, Vol. VIII, Academic Press, New York-London, 1959. MR 0106454
C. Kearton, Obstructions to embedding and isotopy in the metastable range, Math. Ann. 243 (1979), no. 2, 103–113. MR 543720, DOI 10.1007/BF01420417
C. Kearton, Attempting to classify knot modules and their Hermitian pairings, Knot theory (Proc. Sem., Plans-sur-Bex, 1977) Lecture Notes in Math., vol. 685, Springer, Berlin, 1978, pp. 227–242. MR 521736
C. Kearton, Presentations of $n$-knots, Trans. Amer. Math. Soc. 202 (1975), 123–140. MR 358795, DOI 10.1090/S0002-9947-1975-0358795-1
C. Kearton, Signatures of knots and the free differential calculus, Quart. J. Math. Oxford Ser. (2) 30 (1979), no. 118, 157–182. MR 534830, DOI 10.1093/qmath/30.2.157
C. Kearton, Blanchfield duality and simple knots, Trans. Amer. Math. Soc. 202 (1975), 141–160. MR 358796, DOI 10.1090/S0002-9947-1975-0358796-3
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
C. Kearton and W. B. R. Lickorish, Piecewise linear critical levels and collapsing, Trans. Amer. Math. Soc. 170 (1972), 415–424. MR 310899, DOI 10.1090/S0002-9947-1972-0310899-2
Michel A. Kervaire, Les nœuds de dimensions supérieures, Bull. Soc. Math. France 93 (1965), 225–271 (French). MR 189052
Sadayoshi Kojima, A classification of some even dimensional fibered knots, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), no. 3, 671–683. MR 645407
Sadayoshi Kojima, Classification of simple knots by Levine pairings, Comment. Math. Helv. 54 (1979), no. 3, 356–367. MR 543336, DOI 10.1007/BF02566280
Jerome Levine, Knot modules. I, Trans. Amer. Math. Soc. 229 (1977), 1–50. MR 461518, DOI 10.1090/S0002-9947-1977-0461518-0
J. Levine, Polynomial invariants of knots of codimension two, Ann. of Math. (2) 84 (1966), 537–554. MR 200922, DOI 10.2307/1970459
J. Levine, An algebraic classification of some knots of codimension two, Comment. Math. Helv. 45 (1970), 185–198. MR 266226, DOI 10.1007/BF02567325
Saunders Mac Lane, Homology, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1975 edition. MR 1344215
John Milnor, Introduction to algebraic $K$-theory, Annals of Mathematics Studies, No. 72, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971. MR 0349811
D. G. Northcott, An introduction to homological algebra, Cambridge University Press, New York, 1960. MR 0118752