On diffeomorphisms over 𝑇²-knots

Hirose, Susumu

doi:10.1090/S0002-9939-1993-1155598-1

On diffeomorphisms over $T^ 2$-knots
HTML articles powered by AMS MathViewer

by Susumu Hirose PDF

Proc. Amer. Math. Soc. 119 (1993), 1009-1018 Request permission

Abstract:

For a spun ${T^2}$-knot $({S^4},S(k))$ or a twisted spun ${T^2}$-knot $({S^4},\tilde S(k))$ of a nontrivial knot $k$ in ${S^3}$, there are infinitely many isotopy classes of embeddings of a $2$-torus into a $4$-sphere which have this ${T^2}$-knot as their image. This is shown by solving the following question: Which isotopy classes of diffeomorphisms of $S(k)$ or $({S^4},\tilde S(k))$ have orientation-preserving diffeomorphisms of ${S^4}$ as their extension?

References

Richard Hartley, Identifying noninvertible knots, Topology 22 (1983), no. 2, 137–145. MR 683753, DOI 10.1016/0040-9383(83)90024-1
Zyun’iti Iwase, Good torus fibrations with twin singular fibers, Japan. J. Math. (N.S.) 10 (1984), no. 2, 321–352. MR 884423, DOI 10.4099/math1924.10.321
Zyun’iti Iwase, Dehn-surgery along a torus $T^2$-knot, Pacific J. Math. 133 (1988), no. 2, 289–299. MR 941924, DOI 10.2140/pjm.1988.133.289
Zjuñici Iwase, Dehn surgery along a torus $T^2$-knot. II, Japan. J. Math. (N.S.) 16 (1990), no. 2, 171–196. MR 1091159, DOI 10.4099/math1924.16.171
José M. Montesinos, On twins in the four-sphere. I, Quart. J. Math. Oxford Ser. (2) 34 (1983), no. 134, 171–199. MR 698205, DOI 10.1093/qmath/34.2.171
H. F. Trotter, Non-invertible knots exist, Topology 2 (1963), 275–280. MR 158395, DOI 10.1016/0040-9383(63)90011-9
Friedhelm Waldhausen, On irreducible $3$-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56–88. MR 224099, DOI 10.2307/1970594