# Tuesday, March 29

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Goal: classification of tight contact structures on lens spaces.

Lens spaces: $L_{p, q} = S^3/C_p$ where the action is $\tv{z_1, z_2} \mapsto \tv{e^{2\pi i\over p}, e^{2\pi iq \over p}}$ which has order $p$.
Note $L_{p, q} \cong L_{p, q'}$ when $q\equiv q'\mod p$, so we can assume $-p<q\leq 0$, so $p/q < -1$.

Some examples:

- $L_{1, q} \cong S^3$
- $L_{2, 1} \cong \RP^3\cong \SO_3(\RR)$.

There is a genus 1 Heegaard splitting.
The double branched cover of a 2-bridge link is a lens space:

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![](figures/2022-03-29_11-31-57.png)

All lens spaces can be generated by genus 1 Heegaard splittings?
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:::{.remark}
$-p/q$ Dehn surgery is equivalent to a sequence of linked unknots with numbers $r_1,\cdots, r_k$.
When can this be done in a way that preserves the contact structure?
Idea: Legendrian surgery, which removes a Legendrian knot and reglues.
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:::{.remark}
Let $L$ be Legendrian and $\nu(L)$ is a standard neighborhood (so standard contact structure).
Then $\bd \nu(L) \cong T^2$ is convex with 2 dividing curves, where "slope" is the contact framing.
For $\tv{\theta, x, y} \in S^1\times \RR^2$, set $\alpha = \dx + y\dtheta$.
Then $\tv{\theta, 0, 0}$ is Legendrian.
When can we extend $\xi$ uniquely across surgery $S^1\times \DD^2$?
Need to attach handles along integer framing (choice of integer in $\pi_1 \SO_2(\RR) \cong \ZZ$ corresponding to trivializing the normal bundle $\nu (K)$ in an embedding).
Need good surgery slopes: $\ts{n}_{n\in \ZZ} \intersect \ts{1\over k}_{k\in \ZZ} = \ts{\pm 1}$, relative to the tb-framing.
So $\mathrm{tb}-1$ is the best framing.:

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![](figures/2022-03-29_12-11-25.png)

Stabilize up to $r_K+1$ on each Legendrian knot.
Fact: yields a Stein fillable thing, implies tight contact structure.
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:::{.remark}
There are $-r_0-1$ ways to perform $-r_0 - 2$ stabilizations.
E.g. for $-r_0-2 = 3$, break into positive and negative stabilizations:

- $(3, 0)$
- $(2, 1)$
- $(1, 2)$
- $(0, 3)$

So there are $\prod_{1\leq i\leq k} (-r_k - 1)$ tight contact structures on $-p/q = \tv{r_0, \cdots, r_k}$.
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