I'm confused about the relationship between strong admissibility and weak admissibility for pointed diagrams in Heegaard Floer theory. For reference, here are Ozsváth-Szàbo's original definitions:



> 
> A pointed Heegaard diagram is called *strongly admissible for the ${Spin}^c$ structure $\mathfrak{s}$* if for every nontrivial periodic domain $\mathcal{D}$ with $$\langle c\_1(\mathfrak{s}),H(\mathcal{D})\rangle=2n \geq 0,$$
>  $\mathcal{D}$ has some coefficient $>n$. A pointed Heegaard diagram is called *weakly admissible for $\mathfrak{s}$* if for each nontrivial periodic domain $\mathcal{D}$ with $$\langle c\_1(\mathfrak{s}),H(\mathcal{D})\rangle =0,$$
>  $\mathcal{D}$ has both positive and negative coefficients. 
> 
> 
> 


Here's a concrete case that puzzles me: If $c\_1(\mathfrak{s})$ is torsion, then it evaluates to zero on *every* homology class. In that case, it seems that a diagram is 


* strongly admissible for $\mathfrak{s}$ if every nontrivial periodic domain $\mathcal{D}$ has some positive coefficient, and
* weakly admissible for $\mathfrak{s}$ if every nontrivial periodic domain $\mathcal{D}$ has both positive and negative coefficients.


If these conditions are to coincide for $c\_1(\mathfrak{s})$ torsion (or at least for "strong" to be stronger), then it seems like any nontrivial periodic domain with positive coefficients must also have negative coefficients. Is this true?



Admissibility in Heegaard Floer, especially with torsion Spin^c structures
Yes, this is correct. It is spelled out explicitly on page 20 of András Juhász' [*A survey of Heegaard Floer homology*](http://arxiv.org/abs/1310.3418).




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**Edit**: it was not clear to me that you were also asking why, for torsion spin$^c$ structures, strong admissibility implies weak admissibility. Here's a possible proof: if $\mathcal D$ is periodic, then so is $-\mathcal D$. Hence, both $\mathcal D$ and $-\mathcal D$ have some positive coefficient, i.e. $\mathcal D$ has both positive and negative coefficients.


