The Grapple action states:
Requirements You have at least one free hand. Your target
cannot be more than one size larger than you.You attempt to grab an opponent with your free hand.
Attempt an Athletics check against their Fortitude DC.
You can also Grapple to keep your hold on a creature you
already grabbed.Critical Success Your opponent is restrained until the end of
your next turn unless you move or your opponent Escapes
(page 470).Success Your opponent is grabbed until the end of your next
turn unless you move or your opponent Escapes.Failure You fail to grab your opponent. If you already had
the opponent grabbed or restrained using a Grapple, those
conditions on that creature end.Critical Failure If you already had the opponent grabbed or
restrained, it breaks free. Your target can either grab you,
as if it succeeded at using the Grapple action against you,
or force you to fall and land prone.
Similarly, Snagging Strike is worded as:
Requirements You have one hand free, and your target is within reach of that hand.
You combine an attack with quick grappling moves to throw an enemy off balance as long
as it stays in your reach. Make a Strike while keeping one hand free. If this
Strike hits, the target is flat-footed until the start of your next turn or until
it’s no longer within the reach of your hand, whichever comes first.
Finally, Combat Grab is worded as:
Requirements You have one hand free, and your target is
within reach of that hand.You swipe at your opponent and grab at them. Make a melee
Strike while keeping one hand free. If the Strike hits, you grab
the target using your free hand. The creature remains grabbed
until the end of your next turn or until it Escapes, whichever
comes first.
All three of these actions seem to be unclear on the "state" of the free hand that is doing the grabbing while the target is afflicted with the Grabbed condition. It is clear that the attacker usually must remain within reach of the target, but it is only implied that the hand remains occupied for the duration and unable to be used for other actions.
The intent with this question is to determine if it is possible to Grapple/Snagging Strike/Combat Grab a target with one action, and then follow up with Double Slice (which has a requirement of "You are wielding two melee weapons, each in a different hand."). The character would be wielding a melee weapon in one hand, and a gauntlet with the free-hand and agile qualities in the other, thus meeting the two melee weapon requirement (as the gauntlet counts as a simple melee weapon, working past the unarmed attack not counting as a weapon), as well as the free hand requirement for the grapple action.
Best Answer
This works, but probably not the way that you want it to.
Let's start by looking at the relevant portion of Double Slice:
You are holding a melee weapon that doesn't require two hands in one hand and you are wearing a gauntlet on the other hand. Now let's look at what qualifies as Wielding:
You are holding the one-handed melee weapon with one hand, and the gauntlet is being worn on your other hand. You have met the requirements for wielding both the one-handed melee weapon and the gauntlet, and thus have met the requirements for Double Slice.
The gauntlet has the Free-Hand trait:
This allows you to use your gauntlet hand for actions that require a free hand, such as Grapple, Snagging Strike, or Combat Grab. However, performing a Double Slice after a Grapple or Combat Grab will end the effects of the previous action, due to this section of the Free-Hand rule:
Grapple states:
(Emphasis mine)
Similarly, the wording for Combat Grab is:
(Emphasis mine)
These highlighted sections clearly indicate that your free hand is now being used or occupied, which prevents you from being able to attack using that hand. You may have met the requirements for Double Slice, but the specific rule that your Free-Hand weapon cannot attack while being used is overriding the more general rule that you can Strike with each of your weapons (in accordance with Pathfinder's Specific Beats General game convention).
In order to be able to use Double Slice, you would have to Release whatever you're holding with your free hand as a free action, ending the effect of the Grapple or Combat Grab.
Your Grapple or Combat Grab would end, but not your Snagging Strike
The Snagging Strike action has no such limitations within its wording, however, which means that you can, by RAW, use it and then Double Slice while keeping its effects. I'd like to imagine that a fighter grabs, tugs, and releases a creature while using this action, using several small grabs that don't rely on keeping a continuously hold. While the creature is within your reach, you can keep it off-balance for the duration.