# Tuesday, April 26

:::{.remark}
Recall that $\PHS^3 = \Sigma(2,3,5)$ has a Stein-fillable (and hence tight) contact structure.
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:::{.theorem title="?"}
The negative $-\Sigma(2,3,5)$ admits no tight contact structures.
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:::{.remark}
Let $S=S^3\smts{\pt_1,\pt_2,\pt_3}$ be a pair of pants and consider $X = S\cross S^1$.
Note $\bd X = T^2\union T^2\union T^2$:

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![](figures/2022-04-26_11-20-51.png)

Note $-\Sigma(2,3,5) = \Sigma(2,-3,-5)$, since $\PHS^3$ is $-1$ surgery on the trefoil.
Glue in 3 solid torii by
\[
A_1 = \matt 2 {-1} 1 0, \quad A_2 = \matt 3 1 {-1} 0, \quad A_3 = \matt 5 1 {-1} 0
.\]
acting on $\tv{m, \lambda}$ in $S^1\times \DD^2$:

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![](figures/2022-04-26_11-24-08.png)

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:::{.exercise title="?"}
Show via Kirby calculus:

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![](figures/2022-04-26_11-25-37.png)

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:::{.lemma title="?"}
There exist Legendrian representatives $F_2, F_3$ with twisting numbers $m_2, m_3 = -1$.
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:::{.proof title="?"}
Idea: by stabilization, we can assume $m_2,m_3 < 0$, and the claim is that we can destabilize them back up to $-1$ simultaneously using bypass moves.
Reduce to studying dividing sets on $T^2\times I$ or $S^1\times \DD^2$.
Check that the dividing set has slope $-1/2$, which implies that there is an overtwisted disc.
Reduce to $2\cdot 3\cdot 5 = 30$ cases, check that an overtwisted disc can be found in each case.
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