Elementary surgery manifolds and the elementary ideals

Neuzil, J. P.

doi:10.1090/S0002-9939-1978-0464216-9

Elementary surgery manifolds and the elementary ideals
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by J. P. Neuzil PDF

Proc. Amer. Math. Soc. 68 (1978), 225-228 Request permission

Abstract:

We prove the following: If ${M^3}$ is a closed 3-manifold obtained by elementary surgery on a knot K in ${S^3}$ and ${H_1}({M^3})$ is a nontrivial cyclic group, then the first elementary ideal ${\pi _1}({M^3})$ in the integral group ring of ${H_1}({M^3})$ is the principal ideal generated by the polynomial of K.

References

Kunio Murasugi, On the genus of the alternating knot. I, II, J. Math. Soc. Japan 10 (1958), 94–105, 235–248. MR 99664, DOI 10.2969/jmsj/01010094
Richard H. Crowell and Ralph H. Fox, Introduction to knot theory, Ginn and Company, Boston, Mass., 1963. Based upon lectures given at Haverford College under the Philips Lecture Program. MR 0146828
Ralph H. Fox, Free differential calculus. III. Subgroups, Ann. of Math. (2) 64 (1956), 407–419. MR 95876, DOI 10.2307/1969592
Louise E. Moser, On the impossibility of obtaining $S^{2}\times S^{1}$ by elementary surgery along a knot, Pacific J. Math. 53 (1974), 519–523. MR 358756
Kunio Murasugi, On the genus of the alternating knot. I, II, J. Math. Soc. Japan 10 (1958), 94–105, 235–248. MR 99664, DOI 10.2969/jmsj/01010094