Why was Robb's sword across his knees?

Question

In Chapter 24 of AGOT, when Robb Stark greets Tyrion Lannister and some men of the Night's Watch:

His sword was across his knees, the steel bare of all the world to see.

Why did he do this?

Answer 1

Per http://www.westeros.org/Citadel/Concordance/Section/2.3./ (10th bullet)

"A lord with a bared sword across his knees is making a traditional sign that he is denying guest right".

So it goes beyond Tyrion being unwelcome, Robb is explicitly signaling that Tyrion will not be allowed to stay and even that Tyrion may not be safe at Winterfell.

After Tyrion presents his plans for Bran's saddle,

Robb Stark finally sheathed his sword. "I may have been hasty with you ... The hospitality of Winterfell is yours if you wish it, Lannister."

(Emphasis mine)

Answer 2

I think it is rather clear from the Wiki page you have cited: "Even Bran knows what it means to greet a guest with an unsheathed sword and can feel the hostility in the room." Robb is making it very clear to Tyrion, that he is not welcome.

On second thought, this could also be a foreshadowing of the events of

the Red Wedding. The Starks are buried in the crypt below Winterfell, and each grave is covered by a statue of the person sitting with a real sword across his lap. This also evokes Bran and Rickon's escape from Winterfell with the swords from the crypt, after the castle is burned.

Answer 3

George R. R. Martin was asked this and explained that he was denying Tyrion guest right.

Another subject was guest rights. He mentioned the time the Tyrion came to Winterfell and Robb met him with bare steel across his lap. That meant that there was no guest rights for Tyrion.
_{So Spake Martin, ARCHON MEETING (OCTOBER 5-7)}

Guest right in the Seven Kingdoms isn't really adhered to too much nowadays except in the North and the true North. It states that no harm shall come to the guest.

One notable custom that the Northmen hold dearer than any other is guest right, the tradition of hospitality by which a man may offer no harm to a guest beneath his roof, nor a guest to his host. The Andals held to something like it as well, but it looms less large in southron minds. In his text Justice and Injustice in the North: Judgments of Three Stark Lords, Maester Egbert notes that crimes in the North in which guest right was violated were rare but were invariably treated as harshly as the direst of treasons. Only kinslaying is deemed as sinful as the violations of these laws of hospitality.
_{The World of Ice and Fire, The North}

So although he's not telling Tyrion he can't stay at Winterfell he his telling him he will not be protected during his stay there.