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Isomorphisms and peripheral structure of knot groups. (English) Zbl 0629.57004

We show that every isomorphism of prime knot groups preserves the peripheral subgroups of the knot groups, and therefore every isomorphism of prime knot groups is induced by a homeomorphism of the knot manifolds by Waldhausen. We also show that a cable knot is invertible if and only if its companion is invertible.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
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References:

[1] Culler, M., Gordon, C., Luecke, J., Shalen, P.: Dehn surgery on knots. Bull. Am. Math. Soc. New Ser.13 (No. 1), 43-45 (1985) · Zbl 0571.57008 · doi:10.1090/S0273-0979-1985-15357-1
[2] Dehn, M.: Die beiden Kleeblattschlingen. Math. Ann.75, 402-413 (1914) · doi:10.1007/BF01563732
[3] Feustel, C., Whitten, W.: Groups and complements of knots. Can. J. Math.30, 1284-1295 (1978) · Zbl 0373.55003 · doi:10.4153/CJM-1978-105-0
[4] Fox, R.: On the complementary domains of a certain pair of inequivalent knots. Nederl. Akad. Wetensch. Proc. Ser. A 55=Indagat. Math.14, 37-40 (1952) · Zbl 0046.16802
[5] Johannson, K.: Homotopy equivalences of 3-manifolds with boundary. Preprint · Zbl 0412.57007
[6] Reidemeister, K.: Knotentheorie. (Ergebnisse der Mathematik, Vol. 1), Berlin Heidelberg New York: Springer 1932; Reprint, New York: Chelsea 1948
[7] Schreier, O.: Über die GruppenA a B b =1. Abh. Math. Sem. Univ. Hamburg3, 167-169 (1924) · JFM 50.0070.01 · doi:10.1007/BF02954621
[8] Seifert, H.: Verschlingungsinvarianten. S.-B. Preuss. Akad. Wiss.26, 811-828 (1933) · JFM 59.1238.02
[9] Swarup, G.: Cable knots in homotopy 3-spheres. Q. J. Math. Oxford (2)31, 97-104 (1980) · Zbl 0453.57003 · doi:10.1093/qmath/31.1.97
[10] Swarup, G.: A remark on cable knots. Bull. London Math. Soc.18 (Part 4), 401-402 (1986) · Zbl 0611.57006 · doi:10.1112/blms/18.4.401
[11] Waldhausen, F.: On irreducible 3-manifolds which are sufficiently large. Ann. Math.87, 56-83 (1968) · Zbl 0157.30603 · doi:10.2307/1970594
[12] Waldhausen, F.: Recent results on sufficiently large 3-manifolds. Proc. Symp. Pure Math.32, 21-37 (1978) · Zbl 0391.57011
[13] Whitten, W.: Knot complements and groups. Preprint · Zbl 0607.57004
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