-curves inducing two different knots with the same -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 with a strong inversion induced by an unknotting tunnel, we have a double covering projection branched over a trivial knot , where is the axis of . Then a set is called a -curve. We construct -curves and the cyclic branched coverings over -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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  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 -manifolds and -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 -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 , in Geometry and Topology (edited by McCrory and T. Shlfrin) Marcel Dekker, New York (1987) 325-346. Zbl0613.57003MR873302

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