Nonsimple, ribbon fibered knots

Miyazaki, Katura

doi:10.1090/S0002-9947-1994-1176509-4

Nonsimple, ribbon fibered knots
HTML articles powered by AMS MathViewer

by Katura Miyazaki PDF

Trans. Amer. Math. Soc. 341 (1994), 1-44 Request permission

Abstract:

The connected sum of an arbitrary knot and its mirror image is a ribbon knot, however the converse is not necessarily true for all ribbon knots. We prove that the converse holds for any ribbon fibered knot which is a connected sum of iterated torus knots, knots with irreducible Alexander polynomials, or cables of such knots. This gives a practical method to detect nonribbon fibered knots. The proof uses a characterization of homotopically ribbon, fibered knots by their monodromies due to Casson and Gordon. We also study when cable fibered knots are ribbon and results which support the following conjecture. Conjecture. If a $(p,q)$ cable of a fibered knot $k$ is ribbon where $p(> 1)$ is the winding number of a cable in ${S^1} \times {D^2}$, then $q = \pm 1$ and $k$ is ribbon.

References

Iain R. Aitchison and Daniel S. Silver, On certain fibred ribbon disc pairs, Trans. Amer. Math. Soc. 306 (1988), no. 2, 529–551. MR 933305, DOI 10.1090/S0002-9947-1988-0933305-5
Francis Bonahon, Cobordism of automorphisms of surfaces, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 237–270. MR 732345
F. Bonahon, Ribbon fibred knots, cobordism of surface diffeomorphisms and pseudo-Anosov diffeomorphisms, Math. Proc. Cambridge Philos. Soc. 94 (1983), no. 2, 235–251. MR 715046, DOI 10.1017/S0305004100061089
A. J. Casson and C. McA. Gordon, A loop theorem for duality spaces and fibred ribbon knots, Invent. Math. 74 (1983), no. 1, 119–137. MR 722728, DOI 10.1007/BF01388533
A. J. Casson and D. D. Long, Algorithmic compression of surface automorphisms, Invent. Math. 81 (1985), no. 2, 295–303. MR 799268, DOI 10.1007/BF01389054
Marc Culler, C. McA. Gordon, J. Luecke, and Peter B. Shalen, Dehn surgery on knots, Ann. of Math. (2) 125 (1987), no. 2, 237–300. MR 881270, DOI 10.2307/1971311
A. L. Edmonds and J. H. Ewing, Remarks on the cobordism group of surface diffeomorphisms, Math. Ann. 259 (1982), no. 4, 497–504. MR 660044, DOI 10.1007/BF01466055
David Eisenbud and Walter Neumann, Three-dimensional link theory and invariants of plane curve singularities, Annals of Mathematics Studies, vol. 110, Princeton University Press, Princeton, NJ, 1985. MR 817982
Murray Gerstenhaber and Oscar S. Rothaus, The solution of sets of equations in groups, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 1531–1533. MR 166296, DOI 10.1073/pnas.48.9.1531
Patrick M. Gilmer, Ribbon concordance and a partial order on $S$-equivalence classes, Topology Appl. 18 (1984), no. 2-3, 313–324. MR 769297, DOI 10.1016/0166-8641(84)90016-6
Robert E. Gompf and Katura Miyazaki, Some well-disguised ribbon knots, Topology Appl. 64 (1995), no. 2, 117–131. MR 1340864, DOI 10.1016/0166-8641(94)00103-A
C. McA. Gordon, Dehn surgery and satellite knots, Trans. Amer. Math. Soc. 275 (1983), no. 2, 687–708. MR 682725, DOI 10.1090/S0002-9947-1983-0682725-0
C. McA. Gordon, Ribbon concordance of knots in the $3$-sphere, Math. Ann. 257 (1981), no. 2, 157–170. MR 634459, DOI 10.1007/BF01458281
C. McA. Gordon, On primitive sets of loops in the boundary of a handlebody, Topology Appl. 27 (1987), no. 3, 285–299. MR 918538, DOI 10.1016/0166-8641(87)90093-9
C. McA. Gordon and R. A. Litherland, On the signature of a link, Invent. Math. 47 (1978), no. 1, 53–69. MR 500905, DOI 10.1007/BF01609479
Jonathan L. Gross and Thomas W. Tucker, Topological graph theory, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., New York, 1987. A Wiley-Interscience Publication. MR 898434
Richard I. Hartley, Invertible amphicheiral knots, Math. Ann. 252 (1979/80), no. 2, 103–109. MR 593625, DOI 10.1007/BF01420117
William H. Jaco and Peter B. Shalen, Seifert fibered spaces in $3$-manifolds, Mem. Amer. Math. Soc. 21 (1979), no. 220, viii+192. MR 539411, DOI 10.1090/memo/0220

Homotopy equivalence of

manifolds with boundary

Akio Kawauchi, On links not cobordant to split links, Topology 19 (1980), no. 4, 321–334. MR 584558, DOI 10.1016/0040-9383(80)90017-8
Rob Kirby, Problems in low dimensional manifold theory, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 273–312. MR 520548
Charles Livingston and Paul Melvin, Algebraic knots are algebraically dependent, Proc. Amer. Math. Soc. 87 (1983), no. 1, 179–180. MR 677257, DOI 10.1090/S0002-9939-1983-0677257-5
Yaichi Shinohara, On the signature of knots and links, Trans. Amer. Math. Soc. 156 (1971), 273–285. MR 275415, DOI 10.1090/S0002-9947-1971-0275415-1
G. A. Swarup, Cable knots in homotopy $3$-spheres, Quart. J. Math. Oxford Ser. (2) 31 (1980), no. 121, 97–104. MR 564747, DOI 10.1093/qmath/31.1.97
Chichen M. Tsau, Isomorphisms and peripheral structure of knot groups, Math. Ann. 282 (1988), no. 2, 343–348. MR 963021, DOI 10.1007/BF01456980
Friedhelm Waldhausen, On irreducible $3$-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56–88. MR 224099, DOI 10.2307/1970594