Construction of high-dimensional knot groups from classical knot groups

Author:
Magnhild Lien

Journal:
Trans. Amer. Math. Soc. **298** (1986), 713-722

MSC:
Primary 57Q45; Secondary 57M25

MathSciNet review:
860388

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Abstract: In this paper we study constructions of high dimensional knot groups from classical knot groups. We study certain homomorphic images of classical knot groups. Specifically, let be a classical knot group and any element in . We are interested in the quotient groups obtained by centralizing , i.e. , and ask whether is itself a knot group.

For certain and we show that can be realized as the group of a knotted -sphere in -space, but is not realizable by a -sphere in -space. By varying , we also obtain quotients that are groups of knotted -spheres in -space, but they cannot be realized as the groups of classical knots.

We have examples of quotients that have nontrivial second homology. Hence these groups cannot be realized as knot groups of spheres in any dimension. However, we show that these groups are groups of knotted tori in .

**[F]**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****[G]**Tudor Ganea,*Homologie et extensions centrales de groupes*, C. R. Acad. Sci. Paris Sér. A-B**266**(1968), A556–A558 (French). MR**0231914****[H]**Heinz Hopf,*Fundamentalgruppe und zweite Bettische Gruppe*, Comment. Math. Helv.**14**(1942), 257–309 (German). MR**0006510****[J,N]**M. Jankins and W. D. Neumann,*Seifert manifolds*, Lecture notes available from Brandeis University, 1981.**[K]**Michel A. Kervaire,*On higher dimensional knots*, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N.J., 1965, pp. 105–119. MR**0178475****[L]**R. A. Litherland,*Deforming twist-spun knots*, Trans. Amer. Math. Soc.**250**(1979), 311–331. MR**530058**, 10.1090/S0002-9947-1979-0530058-4**[M]**Kunio Murasugi,*On a group that cannot be the group of a 2-knot*, Proc. Amer. Math. Soc.**64**(1977), no. 1, 154–156. MR**0440530**, 10.1090/S0002-9939-1977-0440530-7**[M,K,S]**Wilhelm Magnus, Abraham Karrass, and Donald Solitar,*Combinatorial group theory*, 2nd ed., Dover Publications, Inc., Mineola, NY, 2004. Presentations of groups in terms of generators and relations. MR**2109550****[R]**Joseph J. Rotman,*An introduction to homological algebra*, Pure and Applied Mathematics, vol. 85, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR**538169****[S]**Peter Scott,*The geometries of 3-manifolds*, Bull. London Math. Soc.**15**(1983), no. 5, 401–487. MR**705527**, 10.1112/blms/15.5.401**[S]**John Stallings,*Homology and central series of groups*, J. Algebra**2**(1965), 170–181. MR**0175956****[W]**Edwin Weiss,*Cohomology of groups*, Pure and Applied Mathematics, Vol. 34, Academic Press, New York-London, 1969. MR**0263900****[Y]**Takeshi Yajima,*On a characterization of knot groups of some spheres in 𝑅⁴*, Osaka J. Math.**6**(1969), 435–446. MR**0259893****[Z]**E. C. Zeeman,*Twisting spun knots*, Trans. Amer. Math. Soc.**115**(1965), 471–495. MR**0195085**, 10.1090/S0002-9947-1965-0195085-8

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DOI:
https://doi.org/10.1090/S0002-9947-1986-0860388-1

Article copyright:
© Copyright 1986
American Mathematical Society