I have two questions:


**Question 1.** Suppose that $K$ is a knot in $S^3$. Let $\Sigma(K)$ be the double branched cover of $S^3$ branched along $K$. If $\Sigma(K)=\#\_{i=1}^n\Sigma(2,3,5)$, then $K=\#\_{i=1}^nT\_{3,5}$? 


**Question 2.** There is a conjecture of Ozsvath-Szabo that the only $L$-spaces which are integral homology 3-spheres are connected sums of Poincare homology 3-spheres. If this conjecture and Question 1 is true, then can we (re)prove that Khovanov homology is an unknot-detector using Ozsavth-Szabo's spectral sequence?



Questions on poincare homology spheres and branched covers
As Ian Agol points out the Orbifold Theorem will answer question 1 affirmatively. Although the original result is due to Thurston, the common references in the literature are: 


Boileau, Michel, Sylvain Maillot, and Joan Porti. Three-dimensional orbifolds and their geometric structures. Vol. 15. Paris: Société mathématique de France, 2003.


and


Cooper, Daryl, Craig David Hodgson, and Steve Kerckhoff. Three-dimensional orbifolds and cone-manifolds. Vol. 5. Mathematical Society of Japan, 2000.


However, both of these results would still require one to understand the quotients of $\Sigma(2,3,5)$ by its order 2 symmetries. Dunbar provides such a classification, which serves a translation of Threlfall and Seifert's work, see:


Dunbar, William D. "Geometric orbifolds." Revista Matematica de la Universidad Complutense de Madrid 1.1 (1988): 67-100. (available [here](http://www.mat.ucm.es/serv/revista/vol1-123/vol1-123f.pdf))


and 


Threlfall, William, and Herbert Seifert. "Topologische Untersuchung der Diskontinuitätsbereiche endlicher Bewegungsgruppen des dreidimensionalen sphärischen Raumes I and II." Mathematische Annalen 104 and 107 (1931 and 1932): 1-70 and 543-586.


More specifically Dunbar gives a proof that if a knot $L$ has the property that $\Sigma(L)=\Sigma(2,3,5)$ then $L=T\_{3,5}.$ 


