Why doesn't Borden know which knot he tied?

First of all, as we learn at the end, the whole diary was made up to lead Angier on a wrong track. This means we cannot rely on the fact that he didn't know which knot it was to be the ultimate truth just because it was in his diary (and even less if he just tells him from his own mouth).

So there are different possibilities and I (as well as the movie, I think) cannot give you a definite answer:

1. The statement that he didn't know which knot was a lie and he really knew it. He might have written that into the diary to confuse Angier even further or just to not confess his guilt to him, having chosen the wrong knot (or rather the one Julia wanted, as denoted by the nod she gives him).

2. Like iandotkelly says in his comment, it was indeed always the other twin who told this to Angier and he never got told by his brother about which knot it was because that twin didn't want his brother to know it either (maybe for the same reasons of not confessing his guilt).

3. He maybe really didn't know which knot it was in the sense that he suppressed the memory in order to not even confess his possible guilt to himself and not having to face his responsiblity for Julia's death.

As @ChristianRau suggests, this could simply be guilt, or suppressed memory. I think it is possibly both, coupled with the fact that Borden/Fallon are twins. From the voice over as Angier reads Borden's diary:

BORDEN (V.O.) How often I've fought with my self over that night .. one half of me swearing blind that I tied a simple slip knot... the other half convinced that I tied the Langford double. I suppose I'll never know for sure.

It seems clear from how this is written that one of the twins believed that a simple slip knot had been used, the other believed that a Langford double had been used. It seems likely that the twin responsible believed he'd tied the usual knot, but perhaps suppressed the real memory as @ChristianRau suggests. The other twin might understandably believe, as Angier did, that he must have tied the more secure double. This is supposition, but seems a more plausible than the other way around.

So when Angier asks Borden directly, during the funeral, or when performing the bullet-catch - it seems likely that whatever twin is asked, he just was not sure - even the twin that was Borden that night might have doubts even if he started out convinced he had tied the slip knot. This would be even more likely if by coincidence Angier asks the twin who was Fallon that night, so would really not know for sure.

This is how the Director explained it:

Let's call the first twin "Borden" who loved Olivia (Scarlett Johansson) and was hanged in end, and the other twin "Fallon" who loved Sarah and killed Angier.

It was Borden who tied Julia. He was tieing simple knot then Julia insisted and he tied a Langford Double. Both twins were more curious and smarter than Angier. He did it because he wanted to do something that others couldn't (not to kill Julia as some people suggested)

It was Fallon who wrote the diary, the one at Funeral and same person who got shot at play by Angier. He said that he didn't know which knot he tied because he never asked Borden, because he didn't want to know the truth (or maybe he knew the truth).

Borden's character was less mature in the whole movie. He tied Langford Doubles even though Cutter (Michael Caine) warned him not to, he had an affair with Olivia even though he was married, he went to Angier's play even though his brother told him to leave him to his own damn trick.

The only question worth asking is,"is it possible that Angier encountered the same twin every time he asked which knot he tied". I would say the story goes where the writer points.

It's possible that one of the twins deliberately tied the Langford double knot, knowing Julia won't be able to break free from it in time, in order to sabotage Angier's act and eliminate the business competition - so that the only great magicians remaining in town would be the Borden brothers.

This ruthlessness in business, this willingness to sacrifice both human and animal life for the sake of fame and profit, is paralleled both in the story of Edison's men destroying the fruits of Tesla's hard work, and thus eliminating the competition, and in Angier's own story of destroying himself by drowning, over and over in agony, for the sake of putting on a sensational show. If these men were willing to sacrifice the lives of beautiful birds, their fingers, the happiness of their romantic partners, their own happiness, and even their own lives to sabotage each other's acts and prove the more successful in this business of creating illusions, why wouldn't at least one of the Bordens have it in him to sacrifice the life of an innocent and beautiful woman and the happiness of his business rival in order to nip his competition's success in the bud and come out on top.

Maybe this Borden was a plant from the start, wishing to work with Angier in order to ruin him, just as Olivia was a plant sent to work with Borden in order to ruin him. Maybe he tied the wrong intentionally, after having tested it on his own and determined that it is not possible for anyone to wiggle out of it in time, and maybe he convinced Julia beforehand to be daring and not to protest when he ties it. We can't discount this possibility. This whole movie is about the ruthlessness of business, about how vicious people can become towards one another and how cruel even towards their own selves when they value business success, fame and profit more than life itself.

I think the Borden twin that answered the question the first time genuinely didn't know. I'm sure he found out later or knew already... we ALL knew... BUT anyways he had to COMMIT to his answer for the remainder of his life. That was the price of living the act.

I think the diary entry was written by both the twins at different times because when Angier gets jealous looking at Borden's perfect family life, he says that he found out later in the diary that Borden loved it and hated it the next moment. Since the twins were living a half of both lives, it makes sense that both of them wrote the diary whenever they were Borden.

The scene in which the wrong knot is tied, leading to Julia's death, comes after Borden's claim that he has a new trick in mind (which involved the twin brother). Maybe that's why the other twin(Olivia's Borden) is the one who tied that knot and he is also the one who is asked that question each time by Angier. We can reason that as following:

1. the funeral The other twin is guilty and hence he attends the funeral

2. when Angier tries to kill him at his show performing his gun trick The scene in which Borden tells Sarah about the gun trick is when she tells him he doesn't love her today (which means he is the other twin) This is directly followed by the show scene in which he says that he doesn't know which knot it was.

3. The diary Right after the twin is shot in his hand, he says that he told the truth. 'One half' of him says that it was the simple knot, other half says 'langford double'. Perhaps one half is the other twin and the other half is Borden. I say so because Bordon is certain that he knows a lot about knots and it was indeed the langford double which could have led to the death.

I think that's a trick director uses to give you a false clue. Because when I watched I guessed that Borden could have an identity disorder.

Just my guess - Borden was always insecure about his secret brother. Saying "I don't know" was perhaps a simple way of avoiding further questioning which "might" have led to Robert know his big secret. It is possible both Borden and his brother realized the mess they created by not discussing the previous night's discussion of Borden's twin brother with Cutter, Robert and Julia. Julia hinted to put the Langford double which Borden didn't understand because his twin brother didn't mention about it. Now that the mishap occurred, Borden and his twin brother decided to say "I don't know" as a simple way to hide their twin identity.

Because he's not the one who tied it.

With Borden, there's "the nice one" and "the smart one":

• "Alfred" = the nice one = the one who loved Sarah
• "Freddie" = the smart one = the one who loved Scarlett Johansson

Freddie operates as the "ingenieur" of the pair. He's aggressive and pugnacious. He's the one who wants to tie the Langford Double; he's the one who gets argumentative with Angier, saying "Oh you think you know knots better than me? Do ya?" And he's the one who ties Julia's arms the night she dies.

Alfred is the one with enough empathy to go to the funeral. Angier asks him which knot he tied. Alfred says he doesn't know. He really doesn't! He's not the one who tied it.

This tells us so much about the two brothers. Remember Alfred says "I've asked myself that question a thousand times." When you first hear that, it sounds like a cliche description of soul-searching. But he means it very literally. Borden has asked himself that a thousand times; or rather, Alfred has asked Freddie a thousand times. And the only way Alfred could "not know", is if Freddie insists he tied the simple slip knot, and Alfred doesn't believe him. If Freddie said he'd tied the Langford Double, Alfred would know the answer to Angier's question. But he doesn't.

So Alfred thinks Freddie is a liar. Would lie even to Alfred. That's pretty remarkable.

That leads to another idea: at least part of the diary is genuine. Angier reads it and observes Borden's "divided mind": that suggests both men taking turns writing in the diary. I think Borden took an existing diary, and either EXTENDED it with fake entries up to the point where Scarlett "steals" it; or re-copied it over with the Tesla cipher and fake entries toward the end. Creating a diary from scratch would take a lot of work: easier to use existing entries.
(By the way, if part of the diary is genuine, then Borden is exposing his "divided mind" to Angier, betting that Angier won't understand the secret.)

When Borden ties the knot on Juiia the night she drowns, it's Freddie performing. He shares that significant glance with Julua, which tells us that he ties the Langford Double. And she wasn't able to untie it, as Michael Caine had predicted.

Alfred goes to the funeral.

When Angier goes to Borden's magic show. Alfred is performing that night. Angier holds the gun on him: he still doesn't know, doesn't have any better answer for Angier. So Angier shoots him. This is kind of a cruel irony: the one who didn't do it is shot as punishment. But that means Freddie has to bite down on the stick and take the maiming in cold blood (it's Freddie whose fingers are "still bleeding").

Nazgul's answer here about who's who is right on the money, as I see it, though I differ with him in that I think the twins discussed it a lot. They fought about it. We already know they argue: we see the twins fight & yell at each other once, in the scene where they can't figure out how how Angier does "The Real Transported Man". When Alfred tells us he's "asked himself" a thousand times which knot he tied, those discussions wouldn't have been any less heated.

Nazgul also observes that Freddie is less mature than Alfred, and I think that's important. Freddie ties the Langford; Alfred gets shot in consequence. Freddie starts the affair with Scarlett; Alfred loses his wife in consequence.** Freddie is the one who can't leave Angier alone, and goes below stage at the final performance – but this time Freddie suffers the consequence, gets hanged.

So anyway: that's it. The smart one ("Freddie") tied the knot; the nice one ("Alfred") faced the questions. Offscreen they had many an argument about it, going over the events again and again. Freddie always insisted he tied the right knot, the simple slip. Alfred didn't believe him.

**The way I wrote that sentence minimizes Sarah. As if it's more important that Alfred loses someone than it is that Sarah dies. That's not my view, I'm just going for the parallel sentence construction here, for this point about Freddie compared with Alfred.