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Notes

  1. This is called ambient isotopy in the literature.

  2. The equivalence relation here is ambient isotopy.

  3. The precise notion of equivalence here is again ambient isotopy.

  4. See also a framed knots movie representation available at http://www.youtube.com/watch?v=KxEBhD0C2Pw.

  5. A linking number movie can be viewed at https://www.youtube.com/watch?v=_qOl_5KcANE.

  6. A Seifert surface movie can be viewed at http://www.youtube.com/watch?v=px3Gq_gvvac.

  7. This is a result of Smale’s theorem, which states that every orientation-preserving diffeomorphism of the 2-sphere is smoothly isotopic to the identity map.

References

  1. Colin C. Adams. The Knot Book. American Mathematical Society, 2004. Revised reprint of the 1994 original.

  2. R. H. Bing. Necessary and sufficient conditions that a 3-manifold be \(S^3\). Ann. of Math. 68 (1958), 17–37.

  3. Hansjörg Geiges. An Introduction to Contact Topology. Cambridge Studies in Advanced Mathematics, vol. 109. Cambridge University Press, 2008.

  4. Vaughan F. R. Jones. A Polynomial Invariant for Knots via von Neumann Algebras. Fields Medallists’ Lectures 1997, 448–458.

  5. Louis H. Kauffman. An invariant of regular isotopy. Trans. Amer. Math. Soc. 318:2 (1990), 417–471.

  6. Robion Kirby. A calculus for framed links in \(S^3\). Invent. Math. 45:1 (1978), 35–56.

  7. W. B. R. Lickorish. A representation of orientable combinatorial 3-manifolds. Ann. of Math. (2) 76 (1962), 531–540.

  8. W. B. R. Lickorish. An Introduction to Knot Theory. Graduate Texts in Mathematics, vol. 175. Springer, 1997.

  9. H. R. Morton. Knots, satellites and quantum groups. In Introductory Lectures on Knot Theory, Ser. Knots Everything, vol. 46, pp. 379–406. World Sci., 2012.

  10. James R. Munkres. Topology: A First Course. Prentice-Hall, 1975.

  11. Tomotada Ohtsuki. Quantum Invariants. Series on Knots and Everything, vol. 29, World Scientific, 2002.

  12. Viktor Vasilevich Prasolov and Aleksey B. Sossinsky. Knots, Links, Braids and 3-Manifolds: An Introduction to the New Invariants in Low-Dimensional Topology. American Mathematical Soc., 1997.

  13. K. Reidemeister. Knotentheorie. Springer, 1974.

  14. Dale Rolfsen. Knots and Links. Mathematics Lecture Series, vol. 7. Corrected reprint of the 1976 original. Publish or Perish, 1990.

  15. H. Seifert. Über das Geschlecht von Knoten. Math. Ann. 110:1 (1935), 571–592.

  16. Andrew H. Wallace. Modifications and cobounding manifolds. Canadian J. Math. 12 (1960), 503–528.

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Acknowledgments

The authors would like to thank managing editor David Kramer and the referee for fruitful comments that improved the paper.

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Authors and Affiliations

  1. Department of Mathematics, University of South Florida, Tampa, FL, 33620, USA

    Mohamed Elhamdadi

  2. Department of Computer Science, Ohio State University, Columbus, OH, 43210, USA

    Mustafa Hajij

  3. Columbus, OH, 43210, USA

    Kyle Istvan

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  1. Mohamed Elhamdadi
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Correspondence to Mohamed Elhamdadi.

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Elhamdadi, M., Hajij, M. & Istvan, K. Framed Knots. Math Intelligencer 42, 7–22 (2020). https://doi.org/10.1007/s00283-020-09990-0

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