On certain fibred ribbon disc pairs

Aitchison, Iain R.; Silver, Daniel S.

doi:10.1090/S0002-9947-1988-0933305-5

On certain fibred ribbon disc pairs
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by Iain R. Aitchison and Daniel S. Silver

Trans. Amer. Math. Soc. 306 (1988), 529-551

DOI: https://doi.org/10.1090/S0002-9947-1988-0933305-5

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Abstract:

We prove that for any free group automorphism ${\phi ^{\ast }}$ having a specified form there exists an invertible ribbon disc pair $({B^4}, {D^2})$ such that the closure of ${B_4} - \operatorname {nbd} ({D^2})$ fibres over the circle with fibre a handlebody and monodromy equal to ${\phi ^{\ast }}$. We apply this to obtain results about ribbon $1$- and $2$-knots.

References

Isotoping and twisting knots and ribbons

I. R. Aitchison and J. H. Rubinstein, Fibered knots and involutions on homotopy spheres, Four-manifold theory (Durham, N.H., 1982) Contemp. Math., vol. 35, Amer. Math. Soc., Providence, RI, 1984, pp. 1–74. MR 780575, DOI 10.1090/conm/035/780575
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
Selman Akbulut and Robion Kirby, An exotic involution of $S^{4}$, Topology 18 (1979), no. 1, 75–81. MR 528237, DOI 10.1016/0040-9383(79)90015-6
J. J. Andrews and M. L. Curtis, Free groups and handlebodies, Proc. Amer. Math. Soc. 16 (1965), 192–195. MR 173241, DOI 10.1090/S0002-9939-1965-0173241-8
Kouhei Asano and Katsuyuki Yoshikawa, On polynomial invariants of fibered $2$-knots, Pacific J. Math. 97 (1981), no. 2, 267–269. MR 641156
Richard C. Blanchfield, Intersection theory of manifolds with operators with applications to knot theory, Ann. of Math. (2) 65 (1957), 340–356. MR 85512, DOI 10.2307/1969966
Gerhard Burde, Alexanderpolynome Neuwirthscher Knoten, Topology 5 (1966), 321–330 (German). MR 199858, DOI 10.1016/0040-9383(66)90023-1
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
Tim Cochran, Ribbon knots in $S^{4}$, J. London Math. Soc. (2) 28 (1983), no. 3, 563–576. MR 724727, DOI 10.1112/jlms/s2-28.3.563
Ralph H. Fox and John W. Milnor, Singularities of $2$-spheres in $4$-space and cobordism of knots, Osaka Math. J. 3 (1966), 257–267. MR 211392

Ribbon concordance and a partial order on

equivalence classes

C. McA. Gordon, Some aspects of classical knot theory, Knot theory (Proc. Sem., Plans-sur-Bex, 1977) Lecture Notes in Math., vol. 685, Springer, Berlin, 1978, pp. 1–60. MR 521730
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

Handlebody presentations of knot cobordisms

Taizo Kanenobu, Module d’Alexander des nœuds fibrés et polynôme de Hosokawa des lacements fibrés, Math. Sem. Notes Kobe Univ. 9 (1981), no. 1, 75–84 (French). MR 633997
M. Kervaire and C. Weber, A survey of multidimensional knots, Knot theory (Proc. Sem., Plans-sur-Bex, 1977) Lecture Notes in Math., vol. 685, Springer, Berlin, 1978, pp. 61–134. MR 521731
J. Levine, Some results on higher dimensional knot groups, Knot theory (Proc. Sem., Plans-sur-Bex, 1977) Lecture Notes in Math., vol. 685, Springer, Berlin, 1978, pp. 243–273. With an appendix by Claude Weber. MR 521737
Yoshihiko Marumoto, On ribbon $2$-knots of $1$-fusion, Math. Sem. Notes Kobe Univ. 5 (1977), no. 1, 59–68. MR 467763
John W. Milnor, Infinite cyclic coverings, Conference on the Topology of Manifolds (Michigan State Univ., E. Lansing, Mich., 1967) Prindle, Weber & Schmidt, Boston, Mass., 1968, pp. 115–133. MR 0242163
Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory, Second revised edition, Dover Publications, Inc., New York, 1976. Presentations of groups in terms of generators and relations. MR 0422434
José María Montesinos, Heegaard diagrams for closed $4$-manifolds, Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977) Academic Press, New York-London, 1979, pp. 219–237. MR 537732

Polynôme d’Alexander des noeuds fibres

289

Quach thi Câm Vân and Claude Weber, Une famille infinie de noeuds fibrés cobordants à zéro et ayant même polynôme d’Alexander, Comment. Math. Helv. 54 (1979), no. 4, 562–566 (French). MR 552676, DOI 10.1007/BF02566292
Elvira Strasser Rapaport, Note on Nielsen transformations, Proc. Amer. Math. Soc. 10 (1959), 228–235. MR 104724, DOI 10.1090/S0002-9939-1959-0104724-1
Elvira Rapaport Strasser, Knot-like groups, Knots, groups, and $3$-manifolds (Papers dedicated to the memory of R. H. Fox), Ann. of Math. Studies, No. 84, Princeton Univ. Press, Princeton, N.J., 1975, pp. 119–133. MR 0440531
John R. Stallings, Constructions of fibred knots and links, 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. 55–60. MR 520522
D. W. Sumners, Invertible knot cobordisms, Comment. Math. Helv. 46 (1971), 240–256. MR 290351, DOI 10.1007/BF02566842
Olga Taussky, On a theorem of Latimer and MacDuffee, Canad. J. Math. 1 (1949), 300–302. MR 30491, DOI 10.4153/cjm-1949-026-1
Hidetaka Terasaka, On null-equivalent knots, Osaka Math. J. 11 (1959), 95–113. MR 117736
Takaaki Yanagawa, On ribbon $2$-knots. The $3$-manifold bounded by the $2$-knots, Osaka Math. J. 6 (1969), 447–464. MR 266193
Katsuyuki Yoshikawa, On fibering a class of $n$-knots, Math. Sem. Notes Kobe Univ. 9 (1981), no. 1, 241–245. MR 634013
E. C. Zeeman, Twisting spun knots, Trans. Amer. Math. Soc. 115 (1965), 471–495. MR 195085, DOI 10.1090/S0002-9947-1965-0195085-8