Czechoslovak Mathematical Journal, Vol. 73, No. 2, pp. 603-611, 2023
A note on rational surgeries on a Hopf link
Velibor Bojković, Jovana Nikolić, Mladen Zekić
Received April 4, 2022. Published online February 6, 2023.
Abstract: It is clear that every rational surgery on a Hopf link in 3-sphere is a lens space surgery. In this note we give an explicit computation which lens space is a resulting manifold. The main tool we use is the calculus of continued fractions. As a corollary, we recover the (well-known) result on the criterion for when rational surgery on a Hopf link gives the 3-sphere.
References: [1] T. D. Cochran, R. E. Gompf: Applications of Donaldson's theorems to classical knot concordance, homology 3-spheres and property $P$. Topology 27 (1988), 495-512. DOI 10.1016/0040-9383(88)90028-6 | MR 0976591 | Zbl 0669.57003
[2] R. E. Gompf, A. I. Stipsicz: 4-Manifolfs and Kirby Calculus. Graduate Studies in Mathematics 20. AMS, Providence (1999). DOI 10.1090/gsm/020 | MR 1707327 | Zbl 0933.57020
[3] A. Y. Khinchin: Continued Fractions. The University of Chicago Press, Chicago (1964). MR 0161833 | Zbl 0117.28601
[4] R. Kirby: A calculus for framed links in $S^3$. Invent. Math. 45 (1978), 35-56. DOI 10.1007/BF01406222 | MR 0467753 | Zbl 0377.55001
[5] V. V. Prasolov, A. B. Sossinsky: Knots, Links, Braids and 3-Manifolds: An Introduction to the New Invariants in Low-Dimensional Topology. Translations of Mathematical Monographs 154. AMS, Providence (1997). DOI 10.1090/mmono/154 | MR 1414898 | Zbl 0864.57002
[6] D. Rolfsen: Knots and Links. Mathematical Lecture Series 7. Publish or Perish, Berkeley (1976). MR 0515288 | Zbl 0339.55004
[7] D. Rolfsen: Rational surgery calculus: Extension of Kirby's theorem. Pac. J. Math. 110 (1984), 377-386. DOI 10.2140/pjm.1984.110.377 | MR 0726496 | Zbl 0488.57002
Affiliations: Velibor Bojković, Laboratoire de Mathématiques Nicolas Oresme, 6 Bd Marchal Juin, 14000 Caen, France, e-mail: velibor.bojkovic@unicaen.fr; Jovana Nikolić, University of Belgrade, Faculty of Mathematics, Studentski trg 16, Beograd 105104, Serbia, e-mail: jovanadj@matf.bg.ac.rs; Mladen Zekić (corresponding author), Mathematical Institute of the Serbian Academy of Sciences and Arts, Kneza Mihaila 36, Beograd 11000, Serbia, e-mail: mzekic@mi.sanu.ac.rs
