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 Serie 8 6-B (2003), fasc. n.1, p. 199-209, Unione Matematica Italiana (English)
pdf (537 Kb), djvu (153 Kb). | MR1955705 | Zbl 1150.57002
Sunto
Per un nodo $K$ con un'inversione forte $i$ indotta da un tunnel di scioglimento abbiamo una proiezione $\Pi \colon S^{3}\rightarrow S^{3}/i$ che è un ricoprimento doppio ramificato sopra un nodo banale $\Pi(\text{fix}(i))$, dove $\text{fix}(i)$ è l'asse $i$. Allora un insieme $\Pi(\text{fix}(i)\cup K)$ è chiamato $\theta$-curva. Costruiamo $\theta$-curve e i ricoprimenti $\mathbb{Z}_{2}\oplus \mathbb{Z}_{2}$ ciclici ramificati sopra $\theta$-curve, che hanno due decomposizioni di Heegaard non isotopiche che sono uno stabilmente equivalenti.
Referenze Bibliografiche
[2]
J. S. BIRMAN-
H. M. HILDEN,
Heegaard splittings of branched coverings of $S^3$,
Trans. Amer. Math. Soc.,
270 (
1975), 315-352. |
MR 380765 |
Zbl 0312.55004[3]
A. CASSON-
C. MCA. GORDON,
Reducing Heegaard splittings,
Topology and its applications,
27 (
1987), 273-275. |
MR 918537 |
Zbl 0632.57010[4]
M. J. DUNWOODY,
Cyclic presentations and $3$-manifolds,
Proceedings of Groups- Korea 94' (edited by A. C. Kim and D. L. Johnson) (
1979), 47-55. |
MR 1476948 |
Zbl 0871.20026[5]
Y. H. IM-
S. H. KIM,
Heegaard splittings of the Brieskorn homology spheres that are equivalent after one stabilization,
Note di Mathematica,
20 (1) (
2000/2001), 53-63. |
MR 1885314 |
Zbl 1026.57002[6]
T. KOBAYASHI,
A construction of 3-manifolds whose homeomorphism classes of Heegaard splittings have polynomial growth,
Osaka Journal of Math.,
29 (
1992), 653-674. |
fulltext mini-dml |
MR 1192734 |
Zbl 0785.57005[8]
M. NAKAO,
On the $\mathbb{Z}_2 \oplus \mathbb{Z}_2$ branched coverings of spatial $K_4$-graphs,
Knots 90 (by
Walter de Gruyter), Berlin New York (
1992), 103-116. |
MR 1177415 |
Zbl 0779.57001[9]
E. SEDWICK,
An infinite collection of Heegaard splittings that are equivalent after one stabilization,
Mathematisch Annalen,
308 (
1997), 65-72. |
MR 1446199 |
Zbl 0873.57010[10] H. J. SONG-S. H. KIM, 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]
M. TAKAHASHI,
Two knots with the same $2$-fold branched covering space,
Yokohama Math. J.,
25 (
1977), 91-99. |
MR 461484 |
Zbl 0411.57001[12]
K. WOLCOTT,
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. |
MR 873302 |
Zbl 0613.57003
Con il contributo del Ministero per i Beni e le Attività Culturali