The splittability and triviality of $3$-bridge links
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- by Seiya Negami and Kazuo Okita PDF
- Trans. Amer. Math. Soc. 289 (1985), 253-280 Request permission
Abstract:
A method to simplify $3$-bridge projections of links and knots, called a wave move, is discussed in general situation and it is shown what kind of properties of $3$-bridge links and knots can be recognized from their projections by wave moves. In particular, it will be proved that every $3$-bridge projection of a splittable link or a trivial knot can be transformed into a disconnected one or a hexagon, respectively, by a finite sequence of wave moves. As its translation via the concept of $2$-fold branched coverings of ${S^3}$, it follows that every genus $2$ Heegaard diagram of ${S^2} \times {S^2}\# L(p,q)$ or ${S^3}$ can be transformed into one of specific standard forms by a finite sequence of operations also called wave moves.References
- Joan S. Birman and Hugh M. Hilden, Heegaard splittings of branched coverings of $S^{3}$, Trans. Amer. Math. Soc. 213 (1975), 315–352. MR 380765, DOI 10.1090/S0002-9947-1975-0380765-8
- Wolfgang Haken, Theorie der Normalflächen, Acta Math. 105 (1961), 245–375 (German). MR 141106, DOI 10.1007/BF02559591 C. D. Hodgson, involutions and isotopies of lens spaces, Master Thesis, Univ. of Melbourne.
- Tatsuo Homma and Mitsuyuki Ochiai, On relations of Heegaard diagrams and knots, Math. Sem. Notes Kobe Univ. 6 (1978), no. 2, 383–393. MR 512641
- Tatsuo Homma, Mitsuyuki Ochiai, and Moto-o Takahashi, An algorithm for recognizing $S^{3}$ in $3$-manifolds with Heegaard splittings of genus two, Osaka Math. J. 17 (1980), no. 3, 625–648. MR 591141
- William Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics, vol. 43, American Mathematical Society, Providence, R.I., 1980. MR 565450
- Paik Kee Kim and Jeffrey L. Tollefson, Splitting the PL involutions of nonprime $3$-manifolds, Michigan Math. J. 27 (1980), no. 3, 259–274. MR 584691
- J. Milnor, A unique decomposition theorem for $3$-manifolds, Amer. J. Math. 84 (1962), 1–7. MR 142125, DOI 10.2307/2372800
- José M. Montesinos, Minimal plat representations of prime knots and links are not unique, Canadian J. Math. 28 (1976), no. 1, 161–167. MR 410724, DOI 10.4153/CJM-1976-020-5
- Osamu Morikawa, A counterexample to a conjecture of Whitehead, Math. Sem. Notes Kobe Univ. 8 (1980), no. 2, 295–298. MR 601897
- Seiya Negami, The minimum crossing of $3$-bridge links, Osaka J. Math. 21 (1984), no. 3, 477–487. MR 759476
- Jean-Pierre Otal, Présentations en ponts du nœud trivial, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 16, 553–556 (French, with English summary). MR 679942
- Dale Rolfsen, Knots and links, Mathematics Lecture Series, No. 7, Publish or Perish, Inc., Berkeley, Calif., 1976. MR 0515288
- Horst Schubert, Über eine numerische Knoteninvariante, Math. Z. 61 (1954), 245–288 (German). MR 72483, DOI 10.1007/BF01181346
- Horst Schubert, Knoten mit zwei Brücken, Math. Z. 65 (1956), 133–170 (German). MR 82104, DOI 10.1007/BF01473875
- Jeffrey L. Tollefson, Involutions on $S^{1}\times S^{2}$ and other $3$-manifolds, Trans. Amer. Math. Soc. 183 (1973), 139–152. MR 326738, DOI 10.1090/S0002-9947-1973-0326738-0
- Friedhelm Waldhausen, Über Involutionen der $3$-Sphäre, Topology 8 (1969), 81–91 (German). MR 236916, DOI 10.1016/0040-9383(69)90033-0
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 289 (1985), 253-280
- MSC: Primary 57M25; Secondary 57N10
- DOI: https://doi.org/10.1090/S0002-9947-1985-0779063-6
- MathSciNet review: 779063