An approach to the polygonal knot problem using projections and isotopies

Treybig, L. B.

doi:10.1090/S0002-9947-1971-0279800-3

An approach to the polygonal knot problem using projections and isotopies

Author: L. B. Treybig
Journal: Trans. Amer. Math. Soc. 158 (1971), 409-421
MSC: Primary 55.20
DOI: https://doi.org/10.1090/S0002-9947-1971-0279800-3
MathSciNet review: 0279800
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Abstract: The author extends earlier work of Tait, Gauss, Nagy, and Penney in defining and developing properties of (1) the boundary collection of a knot function, and (2) simple sequences of knot functions or boundary collections. The main results are (1) if two knot functions have isomorphic boundary collections then the knots they determine are equivalent, and (2) if two knot functions determine equivalent knots, then the given functions (their boundary collections) are the ends of a simple sequence of knot functions (boundary collections). Matrices are also defined for knot functions.

[1] R. H. Bing, An alternative proof that 3-manifolds can be triangulated, Ann. of Math. (2) 69 (1959), 37–65. MR 0100841, https://doi.org/10.2307/1970092
[2] Richard H. Crowell and Ralph H. Fox, Introduction to knot theory, Based upon lectures given at Haverford College under the Philips Lecture Program, Ginn and Co., Boston, Mass., 1963. MR 0146828
[3] C. F. Gauss, Werke (8), Teubner, Leipzig, 1900, pp. 272, 282-286.
[4] W. Graeub, Die semilinearen Abbildungen, S.-B. Heidelberger Akad. Wiss. Math.-Nat. Kl. 1950 (1950), 205–272 (German). MR 0042709
[5] Morris L. Marx, The Gauss realizability problem, Proc. Amer. Math. Soc. 22 (1969), 610–613. MR 0244984, https://doi.org/10.1090/S0002-9939-1969-0244984-6
[6] Edwin E. Moise, Affine structures in 3-manifolds. VIII. Invariance of the knot-types; local tame imbedding, Ann. of Math. (2) 59 (1954), 159–170. MR 0061822, https://doi.org/10.2307/1969837
[7] R. L. Moore, Foundations of point set theory, Revised edition. American Mathematical Society Colloquium Publications, Vol. XIII, American Mathematical Society, Providence, R.I., 1962. MR 0150722
[8] Julius Nagy, Über ein topologisches Problem von Gauß, Math. Z. 26 (1927), no. 1, 579–592 (German). MR 1544876, https://doi.org/10.1007/BF01475475
[9] L. P. Neuwirth, Knot groups, Annals of Mathematics Studies, No. 56, Princeton University Press, Princeton, N.J., 1965. MR 0176462
[10] D. E. Penney, An algorithm for establishing isomorphism between tame prime knots in ${E^3}$ , Dissertation, Tulane University, New Orleans, La., 1965.
[11] -, A knot projection has four small complementary domains (unpublished).
[12] D. E. Sanderson, Isotopy in 3-manifolds. I. Isotopic deformations of 2-cells and 3-cells, Proc. Amer. Math. Soc. 8 (1957), 912–922. MR 0090052, https://doi.org/10.1090/S0002-9939-1957-0090052-8
[13] D. E. Sanderson, Isotopy in 3-manifolds. II. Fitting homeomorphisms by isotopy, Duke Math. J. 26 (1959), 387–396. MR 0107231
[14] D. E. Sanderson, Isotopy in 3-manifolds. III. Connectivity of spaces of homeomorphisms, Proc. Amer. Math. Soc. 11 (1960), 171–176. MR 0112128, https://doi.org/10.1090/S0002-9939-1960-0112128-9
[15] P. G. Tait, On knots, Scientific Paper, I, Cambridge Univ. Press, London, 1898.
[16] L. B. Treybig, A characterization of the double point structure of the projection of a polygonal knot in regular position, Trans. Amer. Math. Soc. 130 (1968), 223–247. MR 0217789, https://doi.org/10.1090/S0002-9947-1968-0217789-3
[17] L. B. Treybig, Prime mappings, Trans. Amer. Math. Soc. 130 (1968), 248–253. MR 0217790, https://doi.org/10.1090/S0002-9947-1968-0217790-X