# Questions tagged [contact-topology]

The contact-topology tag has no usage guidance.

13
questions

1
vote

0
answers

71
views

### Almost contact structures on spheres

I will write $(M,\xi)=\mathbf{OB}(P,\phi)$ to denote that $M$ admits an open book decomposition with page $P$ and monodromy $\phi$ supporting a contact structure $\xi$. I will focus on the case where $...

1
vote

0
answers

44
views

### Lagrangian cobordisms from a Legendrian knot to its scaled version

Having a Legendrian knot L in $\mathbb R^3$ and its scale aL (the length of Reeb chords of it are scaled by a>0), are these two Legendrians Legendrian isotopic? Maybe weaker, is there an exact ...

2
votes

2
answers

121
views

### What is the motivation of contact Hamiltonian equation

I've just checked that this is constructed to mimic the ordinary Hamiltonian equation in symplectic geometry. There are several literatures, and they use
$$
\eta(X_H) = -H\\
\mathrm{d}\eta(X_H,-) = \...

2
votes

0
answers

70
views

### Weinstein fillings of a unit cotangent bundle

Given a closed, orientable manifold M, and its unit cotangent bundle $ST^{\ast}M$. I wonder under which conditions $ST^{\ast}M$ admits a subcritical Weinstein filling?

3
votes

0
answers

89
views

### Smooth handle attachment vs Weinstein handle attachment

Given a closed smooth manifold $M$ of dimension $n$, to which we attach a $k$-handle $H_k$.
Take $T^{\ast} M$, can one realize $T^{\ast} (M\cup H_k)$ as a result of symplectic or Weinstein handle ...

4
votes

2
answers

204
views

### Realizing closed manifolds as Legendrian submanifolds of the standard contact vector space

I started learning some basic contact geometry, in particular its flexible side, and I got stuck with the following naive question. Given a closed manifold of dimension $n$, we can always embed it ...

3
votes

0
answers

99
views

### Chekanov-Eliashberg Legendrian DGA with positive grading?

I was just looking back to some notes that I took a few years ago, when I was reading Etnyre's notes on Legendrian Contact Homology in $\mathbb R^3$ and I happened upon the following question that I ...

1
vote

0
answers

46
views

### Linking number of specific Reeb orbits in a toric domain ($S^3$ diffeomorphic)

Consider a toric domain defined by the region bounded on the first quadrant by a function $f:[0,a]\mapsto [0,b]$ with $a,b>0;f(0)=b,f(a)=0,f(x)>0 \hspace{2mm} \forall x\in [0,a)$. We know that $\...

3
votes

1
answer

117
views

### Effect of a Lutz twist on Euler number

I already asked this question on the Math Stack Exchange but did not get an answer.
I am currently working through Geiges proof of the Martinet-Lutz theorem, which can be
found here, and am trying to ...

5
votes

0
answers

68
views

### Transverse open book decompositions supporting the same contact structure

An open book decomposition on an oriented 3-manifold $M$ is a fibered oriented link $B\subset M$, bounding a foliation by Seifert surfaces $\Sigma_t \subset M$, $t\in S^1$, $\partial \Sigma_t = B$.
A ...

2
votes

0
answers

84
views

### How does the Maslov index of a loop `project’ to the rotation number?

I’m trying to learn some Legendrian contact homology and the grading of the generators of the DGA are given by computing a fractional rotation number. In the symplectisation, this number is the Conley-...

1
vote

0
answers

111
views

### Convex surfaces with transverse boundary (contact geometry)

Suppose I have a compact surface $\Sigma$ in a contact 3-manifold, where the boundary $\partial\Sigma$ is transverse to the contact structure. Am I able to perturb $\Sigma$ rel boundary so that it is ...

3
votes

1
answer

115
views

### Transverse knots with knot types of strongly quasi-positive knots

In 2008, Etnyre and Van Horn-Morris proved that if $L$ is a fibered strongly quasi-positive link, there is a unique (up to transverse isotopy) transverse link with the knot type of $L$ in the standard ...