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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?

Edit: For the non-torsion case... Suppose $\mathcal{D}$ is a nontrivial periodic domain. Then $$H(\mathcal{D}):=[\mathcal{D}+\ell \cdot \Sigma+n\cdot D^2],$$ where $n\cdot D^2$ are the attaching disks used to cap off $\mathcal{D}+\ell \cdot \Sigma$. Since $[\Sigma]=0$ in $H_2(Y)$, the class of $H(\mathcal{D})$ equals $[\mathcal{D}+n \cdot D^2]$ and therefore $$H(-\mathcal{D})=[-\mathcal{D}-n \cdot D^2]=-H(\mathcal{D})$$ since the orientation on the boundary components of $-\mathcal{D}$ being reversed requires the attaching disks to have their orientation reversed, too. Then if $\langle c_1(\mathfrak{s}),H(\mathcal{D})\rangle = 2n \geq 0$, we have $$\langle c_1(\mathfrak{s}),H(-\mathcal{D})\rangle = \langle c_1(\mathfrak{s}), -H(\mathcal{D})\rangle = - 2n \leq 0.$$ If $n \neq 0$, then strong admissibility appears to impose no constraints on the coefficients of $-\mathcal{D}$.

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?

I'm confused about the relationship between strong admissibility and weak admissibility for pointed diagrams in Heegaard Floer theory. For reference, here are Ozsvàth-Szábo'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?

Edit: For the non-torsion case... Suppose $\mathcal{D}$ is a nontrivial periodic domain. Then $$H(\mathcal{D}):=[\mathcal{D}+\ell \cdot \Sigma+n\cdot D^2],$$ where $n\cdot D^2$ are the attaching disks used to cap off $\mathcal{D}+\ell \cdot \Sigma$. Since $[\Sigma]=0$ in $H_2(Y)$, the class of $H(\mathcal{D})$ equals $[\mathcal{D}+n \cdot D^2]$ and therefore $$H(-\mathcal{D})=[-\mathcal{D}-n \cdot D^2]=-H(\mathcal{D})$$ since the orientation on the boundary components of $-\mathcal{D}$ being reversed requires the attaching disks to have their orientation reversed, too. Then if $\langle c_1(\mathfrak{s},H(\mathcal{D})\rangle = 2n \geq 0$, we have $$\langle c_1(\mathfrak{s},H(-\mathcal{D})\rangle = \langle c_1(\mathfrak{s}, -H(\mathcal{D})\rangle = - 2n \leq 0.$$ If $n \neq 0$, then strong admissibility appears to impose no constraints on the coefficients of $-\mathcal{D}$.

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?

Edit: For the non-torsion case... Suppose $\mathcal{D}$ is a nontrivial periodic domain. Then $$H(\mathcal{D}):=[\mathcal{D}+\ell \cdot \Sigma+n\cdot D^2],$$ where $n\cdot D^2$ are the attaching disks used to cap off $\mathcal{D}+\ell \cdot \Sigma$. Since $[\Sigma]=0$ in $H_2(Y)$, the class of $H(\mathcal{D})$ equals $[\mathcal{D}+n \cdot D^2]$ and therefore $$H(-\mathcal{D})=[-\mathcal{D}-n \cdot D^2]=-H(\mathcal{D})$$ since the orientation on the boundary components of $-\mathcal{D}$ being reversed requires the attaching disks to have their orientation reversed, too. Then if $\langle c_1(\mathfrak{s}),H(\mathcal{D})\rangle = 2n \geq 0$, we have $$\langle c_1(\mathfrak{s}),H(-\mathcal{D})\rangle = \langle c_1(\mathfrak{s}), -H(\mathcal{D})\rangle = - 2n \leq 0.$$ If $n \neq 0$, then strong admissibility appears to impose no constraints on the coefficients of $-\mathcal{D}$.

I'm confused about the relationship between strong admissibility and weak admissibility for pointed diagrams in Heegaard Floer theory. For reference, here are Ozsvàth-Szábo'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?

I'm confused about the relationship between strong admissibility and weak admissibility for pointed diagrams in Heegaard Floer theory. For reference, here are Ozsvàth-Szábo'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?

Edit: For the non-torsion case... Suppose $\mathcal{D}$ is a nontrivial periodic domain. Then $$H(\mathcal{D}):=[\mathcal{D}+\ell \cdot \Sigma+n\cdot D^2],$$ where $n\cdot D^2$ are the attaching disks used to cap off $\mathcal{D}+\ell \cdot \Sigma$. Since $[\Sigma]=0$ in $H_2(Y)$, the class of $H(\mathcal{D})$ equals $[\mathcal{D}+n \cdot D^2]$ and therefore $$H(-\mathcal{D})=[-\mathcal{D}-n \cdot D^2]=-H(\mathcal{D})$$ since the orientation on the boundary components of $-\mathcal{D}$ being reversed requires the attaching disks to have their orientation reversed, too. Then if $\langle c_1(\mathfrak{s},H(\mathcal{D})\rangle = 2n \geq 0$, we have $$\langle c_1(\mathfrak{s},H(-\mathcal{D})\rangle = \langle c_1(\mathfrak{s}, -H(\mathcal{D})\rangle = - 2n \leq 0.$$ If $n \neq 0$, then strong admissibility appears to impose no constraints on the coefficients of $-\mathcal{D}$.