Knots with infinitely many minimal spanning surfaces

Eisner, Julian R.

doi:10.1090/S0002-9947-1977-0440528-3

Knots with infinitely many minimal spanning surfaces
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by Julian R. Eisner

Trans. Amer. Math. Soc. 229 (1977), 329-349

DOI: https://doi.org/10.1090/S0002-9947-1977-0440528-3

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Addendum: Trans. Amer. Math. Soc. 233 (1977), 367-369.

Abstract:

We show that if ${k_1}$ and ${k_2}$ are nonfibered knots, then the composite knot $K = {k_1}\# {k_2}$ has an infinite collection of minimal spanning surfaces, no two of which are isotopic by an isotopy which leaves the knot K fixed. This result is then applied to show that whether or not a knot has a unique minimal spanning surface can depend on what definition of spanning surface equivalence is used.

References

On the deformation of an n-cell

9

W. R. Alford, Complements of minimal spanning surfaces of knots are not unique, Ann. of Math. (2) 91 (1970), 419–424 (French). MR 253312, DOI 10.2307/1970584
W. R. Alford and C. B. Schaufele, Complements of minimal spanning surfaces of knots are not unique. II. , Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969) Markham, Chicago, Ill., 1970, pp. pp 87–96. MR 0288751
R. H. Bing, Locally tame sets are tame, Ann. of Math. (2) 59 (1954), 145–158. MR 61377, DOI 10.2307/1969836
Gerhard Burde and Heiner Zieschang, Neuwirthsche Knoten und Flächenabbildungen, Abh. Math. Sem. Univ. Hamburg 31 (1967), 239–246 (German). MR 229229, DOI 10.1007/BF02992402

Complements of minimal spanning surfaces

R. J. Daigle, More on complements of minimal spanning surfaces, Rocky Mountain J. Math. 3 (1973), 473–482. MR 367968, DOI 10.1216/RMJ-1973-3-3-473
M. Dehn, Über die Topologie des dreidimensionalen Raumes, Math. Ann. 69 (1910), no. 1, 137–168 (German). MR 1511580, DOI 10.1007/BF01455155
D. B. A. Epstein, Curves on $2$-manifolds and isotopies, Acta Math. 115 (1966), 83–107. MR 214087, DOI 10.1007/BF02392203
R. H. Fox, A quick trip through knot theory, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 120–167. MR 0140099
F. González-Acuña, Dehn’s construction on knots, Bol. Soc. Mat. Mexicana (2) 15 (1970), 58–79. MR 356022
J. M. Kister, Isotopies in $3$-manifolds, Trans. Amer. Math. Soc. 97 (1960), 213–224. MR 120628, DOI 10.1090/S0002-9947-1960-0120628-5
Herbert C. Lyon, Simple knots without unique minimal surfaces, Proc. Amer. Math. Soc. 43 (1974), 449–454. MR 377850, DOI 10.1090/S0002-9939-1974-0377850-8
Herbert C. Lyon, Simple knots with unique spanning surfaces, Topology 13 (1974), 275–279. MR 346770, DOI 10.1016/0040-9383(74)90020-2
Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1966. MR 0207802
Edwin E. Moise, Affine structures in $3$-manifolds. V. The triangulation theorem and Hauptvermutung, Ann. of Math. (2) 56 (1952), 96–114. MR 48805, DOI 10.2307/1969769
L. P. Neuwirth, Knot groups, Annals of Mathematics Studies, No. 56, Princeton University Press, Princeton, N.J., 1965. MR 0176462
C. D. Papakyriakopoulos, On Dehn’s lemma and the asphericity of knots, Ann. of Math. (2) 66 (1957), 1–26. MR 90053, DOI 10.2307/1970113
Joseph J. Rotman, The theory of groups. An introduction, 2nd ed., Allyn and Bacon Series in Advanced Mathematics, Allyn and Bacon, Inc., Boston, Mass., 1973. MR 0442063
Horst Schubert, Die eindeutige Zerlegbarkeit eines Knotens in Primknoten, S.-B. Heidelberger Akad. Wiss. Math.-Nat. Kl. 1949 (1949), no. 3, 57–104 (German). MR 0031733
Horst Schubert, Knoten und Vollringe, Acta Math. 90 (1953), 131–286 (German). MR 72482, DOI 10.1007/BF02392437
Horst Schubert and Kay Soltsien, Isotopie von Flächen in einfachen Knoten, Abh. Math. Sem. Univ. Hamburg 27 (1964), 116–123 (German). MR 166780, DOI 10.1007/BF02993061
H. Seifert, Über das Geschlecht von Knoten, Math. Ann. 110 (1935), no. 1, 571–592 (German). MR 1512955, DOI 10.1007/BF01448044
John Stallings, On fibering certain $3$-manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 95–100. MR 0158375
John Stallings, Group theory and three-dimensional manifolds, Yale Mathematical Monographs, vol. 4, Yale University Press, New Haven, Conn.-London, 1971. A James K. Whittemore Lecture in Mathematics given at Yale University, 1969. MR 0415622

Bounds in piecewise linear topology

The number of isotopy types of knot spanning surfaces of maximal characteristic

20

H. F. Trotter, Some knots spanned by more than one knotted surface of mininal genus, 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. 51–62. MR 0377851
Wilbur Whitten, Isotopy types of knot spanning surfaces, Topology 12 (1973), 373–380. MR 372842, DOI 10.1016/0040-9383(73)90029-3