In Hamlet, how would one characterize the relationship between Claudius and Gertrude in Act 4 Scene 1?Quote lines.

In Shakespeare's Hamlet, the
relationship between Claudius and Gertrude is as strong as ever in Act 4.1--at least on
the surface.

She immediately tells Claudius exactly what he
asks for:  what do her "sighs" and "heaves" mean and where and how is Hamlet (Act
4.1.1-5)? 

She tells him that Hamlet is
as

Mad as the
sea and wind when both contend

Which is the mightier.  (Act
4.1.6-7)

And she informs
Claudius of the details concerning Hamlet's killing of Polonius, and that Hamlet is
removing the corpse from her chamber.

But she doesn't stop
there.  She also tells Claudius that Hamlet "...weeps for what is done" (Act 4.1.27)--is
sorry for killing Polonius. 

Like much in the play,
however, Gertrude's actions are ambiguous, and thus, while the relationship between her
and Claudius appears to be as strong as ever, there is a possibility that it is
not.

Gertrude, in fact, tells Claudius exactly what Hamlet
asks her to tell him just a few minutes before Act
4.1:

[Don't
let Claudius] Make you to ravel all this matter out,

That I
essentially am not in madness,

But mad in craft.  (Act
3.4.190-193)

Hamlet tells her
to not let Claudius sweet talk her into telling the truth about Hamlet's madness--that
he is only acting.  And she doesn't.  She tells Claudius that Hamlet is as mad as the
wind and the sea when they contend to see which is the most powerful.  Then, as
mentioned, she also defends her son and tries to alleviate the seriousness of what he's
done by telling Claudius that Hamlet is sorry for killing
Polonius. 

Her words are ambiguous, because we don't know
her motivation.  She may tell Claudius that Hamlet is mad because Hamlet asks her to, or
she may tell Claudius that Hamlet is mad because she really thinks he
is. 

Gertrude promises Hamlet that she will do as he
asks:

Be thou
assured, if words be made of breath

And breath of life, I
have no life to breathe

What thou hast said to me.  (Act
3.4.200-202)

She will not
speak the truth about Hamlet's madness.  And, again, she
doesn't. 

Yet, after seeing Hamlet kill Polonius, then
speak to the air as if there is someone present (he sees the Ghost, she doesn't), the
possibility certainly exists that she is merely saying what Hamlet wants her to say, and
that she still believes Hamlet is, indeed, mad.  And one could hardly blame
her. 

Thus, the relationship between Gertrude and Claudius
might be as strong as ever in Act 4.1, or a rift may have developed:  Gertrude may have
switched her allegiance from Claudius to her son, Hamlet.