# Monodromy Representation of Elliptic Curve

I’m reading this post by Charles Siegel on Monodromy Representations
and there is a construction in example a not unterstand.

We look at the family $$y^2z=x(x-z)(x-\lambda z)y^2z=x(x-z)(x-\lambda z)$$ of projective elliptic
curves parametrized by $$\lambda\lambda$$.
We’re ignoring degenerations for the moment so this lives over
$$\mathbb{P}^1\setminus\{0,1,\infty\}\mathbb{P}^1\setminus\{0,1,\infty\}$$. This is also visible as
$$\mathbb{C}^\times\setminus\{1\}\mathbb{C}^\times\setminus\{1\}$$.

Since topologically this space
is homotopy equivalent to two-loop space $$S^1 \vee S^1S^1 \vee S^1$$, we have
$$\pi_1(\mathbb{C}^\times\setminus\{1\})= \mathbb{Z} * \mathbb{Z}\pi_1(\mathbb{C}^\times\setminus\{1\})= \mathbb{Z} * \mathbb{Z}$$
we’ve got two nontrivial loops,
one around zero, the other around one, and there are no relations,
this is the free group of rank $$22$$: We can envision all of this as
living in the plane with four points marked as $${0,1,\infty,\lambda}{0,1,\infty,\lambda}$$,
and a cut from $${0}{0}$$ to $${\lambda}{\lambda}$$ and a cut from $${1}{1}$$ to $${\infty}{\infty}$$.
Then the loops are what happens if we loop $${\lambda}{\lambda}$$ around things.
We get one loop from rotating the cut between $${0}{0}$$ and $${\lambda}{\lambda}$$
all the way around. The other one is a bit trickier to describe
directly, but we can describe it in terms like this.

Later we consider the action of the two loops from
$$\pi_1(\mathbb{P}^1 \setminus \{0,1,\infty\})\pi_1(\mathbb{P}^1 \setminus \{0,1,\infty\})$$ on the homology
group $$H_1H_1$$ of the elliptic curve. The generators of $$H_1H_1$$ are: We’ll focus on the first one. Look at $${H_1}{H_1}$$ of the elliptic curve.
It has two generators, call them $${\delta}{\delta}$$ and $${\gamma}{\gamma}$$, where
$${\delta}{\delta}$$ is a loop around the $${0\lambda}{0\lambda}$$ cut, and $${\gamma}{\gamma}$$ is a
loop through the two cuts. This is a standard homology basis,
and we’ll look at the action of our element of
$$\pi_1(\mathbb{P}^1\setminus\{0,1,\infty\})\pi_1(\mathbb{P}^1\setminus\{0,1,\infty\})$$ on the homology:

There are some aspects I not understand in this construction. Firstly, what
does the author mean when he wrote ‘The loops are what happens if we
loop $$\lambda\lambda$$ around things. We get one loop from rotating
the cut between $${0}{0}$$ and $${\lambda}{\lambda}$$ all the way around.’ Which
loops do we obtain by this operation rotating the cut between
$${0}{0}$$ and $${\lambda}{\lambda}$$? I not understand which objects we obtain doing
it. The only involved loops here are the two generators of
$$\pi_1(\mathbb{P}^1\setminus\{0,1,\infty\})\pi_1(\mathbb{P}^1\setminus\{0,1,\infty\})$$ called
$$\gamma_1\gamma_1$$ and $$\gamma_2\gamma_2$$ and the two generators of $$H_1H_1$$ of the
elliptic curve $$\delta\delta$$ and $$\gamma\gamma$$ in second picture, which
as far as I understand the second picture represents one
“sheet” of the elliptic curve. I would like do understand which loops the author has in mind?

Is the which the author has in ‘We get one loop from rotating the cut between
$$00$$ and $$λλ$$ all the way around’ just literally this one: Secondly why the induced action by
$$\gamma_2 \in \pi_1(\mathbb{P}^1\setminus\{0,1,\infty\})\gamma_2 \in \pi_1(\mathbb{P}^1\setminus\{0,1,\infty\})$$ on $$H_1H_1$$
correspond precisely to the rotations of the cut between $${0}{0}$$ and
$${\lambda}{\lambda}$$ discussed in first part? Can somebody explain the geometry of this action?