θ -curves inducing two different knots with the same 2 -fold branched covering spaces

Soo Hwan Kim; Yangkok Kim

Bollettino dell'Unione Matematica Italiana (2003)

  • Volume: 6-B, Issue: 1, page 199-209
  • ISSN: 0392-4041

Abstract

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For a knot K with a strong inversion i induced by an unknotting tunnel, we have a double covering projection Π : S 3 → S 3 / i branched over a trivial knot Π ⁢ fix ⁢ i , where fix ⁢ i is the axis of i . Then a set Π ⁢ fix ⁢ i ∪ K is called a θ -curve. We construct θ -curves and the Z 2 ⊕ Z 2 cyclic branched coverings over θ -curves, having two non-isotopic Heegaard decompositions which are one stable equivalent.

How to cite

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Kim, Soo Hwan, and Kim, Yangkok. "$\theta$-curves inducing two different knots with the same $2$-fold branched covering spaces." Bollettino dell'Unione Matematica Italiana 6-B.1 (2003): 199-209. <http://eudml.org/doc/196041>.

@article{Kim2003,
abstract = {For a knot $K$ with a strong inversion $i$ induced by an unknotting tunnel, we have a double covering projection $\Pi \colon S^\{3\}\rightarrow S^\{3\}/i$ branched over a trivial knot $\Pi(\text\{fix\}(i))$, where $\text\{fix\}(i)$ is the axis of $i$. Then a set $\Pi(\text\{fix\}(i)\cup K)$ is called a $\theta$-curve. We construct $\theta$-curves and the $\mathbb\{Z\}_\{2\}\oplus \mathbb\{Z\}_\{2\}$ cyclic branched coverings over $\theta$-curves, having two non-isotopic Heegaard decompositions which are one stable equivalent.},
author = {Kim, Soo Hwan, Kim, Yangkok},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {199-209},
publisher = {Unione Matematica Italiana},
title = {$\theta$-curves inducing two different knots with the same $2$-fold branched covering spaces},
url = {http://eudml.org/doc/196041},
volume = {6-B},
year = {2003},
}

TY - JOUR
AU - Kim, Soo Hwan
AU - Kim, Yangkok
TI - $\theta$-curves inducing two different knots with the same $2$-fold branched covering spaces
JO - Bollettino dell'Unione Matematica Italiana
DA - 2003/2//
PB - Unione Matematica Italiana
VL - 6-B
IS - 1
SP - 199
EP - 209
AB - For a knot $K$ with a strong inversion $i$ induced by an unknotting tunnel, we have a double covering projection $\Pi \colon S^{3}\rightarrow S^{3}/i$ branched over a trivial knot $\Pi(\text{fix}(i))$, where $\text{fix}(i)$ is the axis of $i$. Then a set $\Pi(\text{fix}(i)\cup K)$ is called a $\theta$-curve. We construct $\theta$-curves and the $\mathbb{Z}_{2}\oplus \mathbb{Z}_{2}$ cyclic branched coverings over $\theta$-curves, having two non-isotopic Heegaard decompositions which are one stable equivalent.
LA - eng
UR - http://eudml.org/doc/196041
ER -

References

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  2. BIRMAN, J. S.- HILDEN, H. M., Heegaard splittings of branched coverings of S 3 , Trans. Amer. Math. Soc., 270 (1975), 315-352. Zbl0312.55004MR380765
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  5. IM, Y. H.- KIM, S. H., Heegaard splittings of the Brieskorn homology spheres that are equivalent after one stabilization, Note di Mathematica, 20 (1) (2000/2001), 53-63. Zbl1026.57002MR1885314
  6. KOBAYASHI, T., A construction of 3-manifolds whose homeomorphism classes of Heegaard splittings have polynomial growth, Osaka Journal of Math., 29 (1992), 653-674. Zbl0785.57005MR1192734
  7. MORIMOTO, K.- SAKUMA, M.- YOKOTA, Y., Identifying tunnel number one knots, J. Math. Soc. Japan, 48 (4) (1996), 667-688. Zbl0869.57008MR1404816
  8. NAKAO, M., On the Z 2 ⊕ Z 2 branched coverings of spatial K 4 -graphs, Knots 90 (by Walter de Gruyter), Berlin New York (1992), 103-116. Zbl0779.57001MR1177415
  9. SEDWICK, E., An infinite collection of Heegaard splittings that are equivalent after one stabilization, Mathematisch Annalen, 308 (1997), 65-72. Zbl0873.57010MR1446199
  10. SONG, H. J.- KIM, S. H., Dunwoody 3 -manifolds and 1 , 1 -decomposiable knots, Proc. Workshop in pure math (edited by Jongsu Kim and Sungbok Hong), Geometry and Topology, 19 (2000), 193-211. 
  11. TAKAHASHI, M., Two knots with the same 2 -fold branched covering space, Yokohama Math. J., 25 (1977), 91-99. Zbl0411.57001MR461484
  12. WOLCOTT, K., The knotting of theta curves and other graphs in S 3 , in Geometry and Topology (edited by McCrory and T. Shlfrin) Marcel Dekker, New York (1987) 325-346. Zbl0613.57003MR873302

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